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CO o GO O cz 03 0Q April 1983 Vol. 65, No. 4 CD CD CO CD "CO CD "O CD 5 Why Do Food Prices Increase? 13 Polynomial Distributed Lags and the Estimation of the St. Louis Equation 26 Weekly Money Supply Forecasts: Effects of the October 1979 Change in Monetary Control Procedures The Review is published 10 times per year by the Research and Public Information D epartment o f the Federal Reserve Rank o f St. Louis. Single-copy subscriptions are available to the public free o f charge. Mail requests fo r subscriptions, back issues, or address changes to: Research and Public Information Department, Federal Reserve Rank o f St. Louis, P.O. Rox 442, St. Louis, Missouri 63166. Articles herein may be reprinted provided the source is credited. Please provide the Rank s Research and Public Information Department with a copy o f reprinted material. Federal Reserve Bank of St. Louis Review April 1983 In This Issue . . . This issue of the Review contains three articles that investigate the influence of changes in money growth and monetary policy actions on diverse economic behavior. In the first article, “Why Do Food Prices Increase?” Michael Belongia discuss es the various explanations that have been offered to account for increases in food prices. Many popular explanations (for example, unionization, price supports and “m iddlem en”) fail to distinguish betw een relative prices and nominal (or money) prices. Taking this distinction into account, the author analyzes graphically the different patterns of price behavior that would be observed under each type of price change. Plots of actual data suggest that most of the recent changes in food prices have followed a path similar to that for changes in the nominal prices of other goods. Therefore, models that explain isolated changes in relative prices are of limited use, at best, in explaining ongoing changes in nominal food prices. A statistical analysis of food prices from 1960 through 1982 shows that the primary cause of changes in the food component of the Consum er Price Index (CPI) has been the past growth of the money stock. Belongia’s analysis thus indicates that, while many of the current explanations are inconsistent with the actual behavior of food prices, the rate of increase in the food component of the CPI in the current quarter shares an approximate one-to-one correspondence with the rate of growth of the money stock over the previous four quarters. In the second article, “Polynomial D istributed Lags and the Estimation of the St. Louis Equation,” Dallas S. Batten and Daniel L. Thornton engage in a detailed re-estimation of the nature of the impact of money growth and government expenditures in the well-known St. Louis equation. The major purpose of the study is to determ ine w hether the conclusions drawn from previous estimations of this equation depend on the selection of lag length or the imposition of polynomial restrictions. In conducting this examination, the authors generalize a procedure for selecting the lag length and polynomial degree that is both convenient and computationally efficient. They find that the St. Louis equation’s policy conclusions are unaffected by the lag length selected or the polynomial restrictions imposed. In particular, the long-run effectiveness of money growth on nominal spending growth and the long-run ineffectiveness of the growth in government spending are substantiated. Their investigation also identifies a different specification of the equation that outperforms the currently used St. Louis equation in terms of both in-sample and out-of-sample criteria. This new specification has substantially longer lags for both money and government spending growth and more polynomial restrictions than the currently specified St. Louis equation. In the third article, R. W. Hafer focuses on the predictions of weekly money growth that financial analysts use in attem pting to anticipate Federal Reserve policy actions. Although several studies have shown the weekly M l num bers to be 3 In This Issue . . . Digitized for 4 FRASER unreliable predictors of long-term policy trends, weekly predictions of M l fre quently are used to determ ine short-term financial market strategies. In “Weekly Money Forecasts,” Hafer examines whether the October 6, 1979, change in the Federal Reserve’s procedures to control the money supply affected the fore casters’ abilities to predict the change in M l. More specifically, he addresses the issue of w hether the change in operating procedures affected the unbiased and efficiency characteristics of these M l forecasts. To answer this question, the author assesses the money supply forecasts from a survey of actively participating money market analysts. Using the average forecast as the “market s’’ prediction, he finds that the change in monetary control proce dures significantly altered the characteristics of the weekly money supply fore casts. Prior to October 1979, forecasts of the weekly change in M l generally were unbiased and efficient estimates of the actual change; since October 1979, these forecasts have been biased and inefficient. These findings, along with those presented in studies that analyze the effects of unanticipated weekly money changes on interest rates, “suggest that a more predictable [monetary policy] control procedure would contribute to a more stable financial m arket.” Why Do Food Prices Increase? MICHAEL T. BELONGIA O v E R the past decade economists have devoted much research effort to identifying factors that in fluence the direction and magnitude of changes in food prices. Under the widely-accepted belief that “food prices rose faster than nonfood prices during the 1970s,” many have attem pted to identify the unique characteristics of food products and their marketing system that have caused food prices to rise faster than the general rate of inflation.1 These studies typically concluded that market concentration and increases in the costs of assorted inputs were the chief causes of increases in retail food prices. Not all analysts share these views, however. First, there is some disagreement concerning w hether food has, in fact, become relatively more expensive in re cent years. Second, recent empirical research has found that increases in food prices are more directly related to the monetary policy of the Federal Reserve than they are related to unique marketing practices of firms in the food industry. Thus, contrary to the pre dom inant view, these argum ents contend that in creases in food prices, on average, share the same path as that followed by other prices. The following discussion attem pts to clarify some of these issues. After several basic economic concepts are defined, a statistical analysis of the data is conducted. The evidence suggests that virtually all of the long-run increases in food prices can be explained by past rates of growth of the money stock. Conversely, the discus sion in the article’s final section indicates that predic- 'See, for example, R. McFall Lamm, “Prices and Concentration in the Food Retailing Industry,” Journal o f Industrial Economics (September 1981), pp. 67-78; Larry E. Salathe and William T. Boehm, Food Prices in Perspective: A Summary Analysis, Eco nomics, Statistics and Cooperatives Service (U.S. D epartm ent of Agriculture 1978); and R. McFall Lamm and Paul C. W escott, “The Effects of Changing Input Costs on Food Prices, ” American Journal o f Agricultural Economics (May 1981), pp. 187-96. tions of competing theories often are contradicted by actual events. RELATIVE VS. NOMINAL PRICES The first step necessary in a discussion of price changes draws the distinction betw een relative and nominal prices. Put most simply, nominal (or money) prices are the actual, dollar-denominated prices at which goods are exchanged; for example, a news paper’s nominal price is 25 cents. A relative price, however, expresses the cost of a good in term s of other goods, not in term s of money. That is, if a book’s nominal price is $2, the relative price of a newspaper — relative to a book — is ’/s($0.25 -r- $2.00 = Vs). This shows that the newspaper is “worth” one-eighth of a book. The importance of this distinction is more than numerical in nature. There is a crucial economic dis tinction betw een nominal and relative prices. Changes in relative prices reflect changes in the rate of exchange betw een goods caused by relative changes in the sup ply and/or dem and for goods; changes in nominal prices reflect changes in the rate of exchange between goods and money associated with changes in the supply and/or demand for money. For example, under a neu tral inflation, in which all nominal (money) prices in crease at the same rate, a 20 percent increase in the price of newspapers to 30 cents would be matched by a 20 percent increase in the price of a book to $2.40 (1.20 X $2.00 = $2.40). This equal percentage increase in all money prices is neutral because relative prices are unaffected; that is, with a neutral 20 percent inflation, the relative price of a newspaper is still Vs ($0.30 h$2.40 = Vs) of the book. The distinguishing feature of an equal percentage change in all nominal prices is that it has no long-run impact on economic activity; that is, it does not change 5 FEDERAL RESERVE BANK OF ST. LOUIS the allocation of resources betw een newspapers and books.2 In other words, when all prices — including incomes — are rising at equal rates, relative prices remain unchanged. In this instance, an individual who allocates fixed proportions of his income to newspa pers, books, food and housing is unaffected by a neutral inflation: even though all prices rise by 10 percent, these changes are offset by a 10 percent increase in income. Nominal price changes of this nature share a one-to-one correspondence with past rates of growth of the money stock.3 Conversely, relative price changes for individual products both result from, and contribute to, changes in economic relationships. For example, if an increase in demand doubled the price of newspapers from 25 cents to 50 cents, an individual who purchased news papers would adjust his spending patterns to reflect this increase. That is, if one person previously had purchased four newspapers per week for $1(4 X $0.25) out of a $100 weekly income, there would be $99 per week to spend on other items. W hen the newspaper price rises to 50 cents, the four newspapers cost $2 and only $98 remains for other purchases. The change in the relative price of newspapers forces this individual to reallocate the $100 of weekly income: either the purchase of newspapers or other goods must be re duced by $1. The issue of changes in food prices also can be re duced to this simple dichotomy betw een movements in relative and nominal prices. Analysts who believe 2Rational expectations theorists may argue that real economic activ ity will be affected in the short run unless price changes are forecast perfectly, e.g., Robert E. Lucas, Jr., “Expectations and the N eu trality of Money , ”Journal o f Economic Theory (April 1972), pp. 103-24. The present analysis also ignores the effects of factors like a progressive tax structure, usury laws and other im pedim ents that prevent or complicate a com plete indexation of this type of price change. For purposes of illustration, however, this simple example is intended only to draw a distinction betw een relative and nominal prices. ‘The linkage betw een past growth rates of the money stock and the current rate of inflation has been established in a num ber of stud ies. Among these are: P eter I. Berman, Inflation and the Money Supply in the United States, 1956-1977 (Lexington Books, 1978); Yash P. Mehra, 'An Empirical Note on Some M onetarist Proposi tions,” Southern Economic Journal (July 1978), pp. 154-67; Robert E. Lucas, “Two Illustrations of the Q uantity Theory of M oney,” American Economic Review (D ecem ber 1980), pp. 1005-14; Denis S. Karnoskv, “The Link Between Money and Prices,” this Review (June 1976), pp. 17-23; Keith M. Carlson, “The Lag from Money to Prices,” this Review (October 1980), pp. 3-10; and John A. Tatom, “Energy Prices and Short-Run Economic Perform ance,” this Re view (January 1981), pp. 3-17. F urther discussion of the distinction betw een inflation and changes in relative prices can be found in Lawrence S. Davidson, “Inflation Misinformation and M onetary Policy,” this Review (June/July 1982), pp. 15-26. Digitized for 6FRASER APRIL 1983 Figure 1 Theoretical Differences Between Rates of Price Change and C hanges in Price Levels N a tu r a l lo garithm s o f p rice Food N o n fo o d 1970 food prices have risen faster than nonfood prices are arguing that shifts in the relative supply and demand conditions for both food and nonfood products have resulted in a net increase in the relative price of food. Conversely, those who argue food prices grew at the same rate as other prices believe that most of the recent changes in food prices can be linked directly to the high rate of money growth that existed over this period. The distinction betw een these views is illus trated in the graphical analysis that follows. ALTERNATIVE INTERPRETATIONS OF HISTORICAL DATA Those who argue that food prices increased at a relatively faster rate than nonfood prices in the 1970s (see footnote 1) base their conclusion on the observa tion that, over this period, the food component of the Consumer Price Index (CPIF) increased by 87 percent compared to a 66 percent increase for the nonfood component (CPINF). Although these statistics are cor rect technically, they are based on total increases for the 10-year period. That is, the 87 percent increase for CPIF is determ ined by constructing the simple differ ence of index values for D ecem ber 1969 and Decem ber 1979. This simple calculation of price change, how ever, fails to distinguish betw een changes in price levels and average rates of price change. To see the problem with this type of calculation, consider figure 1. Lines A, B and C represent different growth paths for the food and nonfood components of FEDERAL RESERVE BANK OF ST. LOUIS the CPI. The horizontal lines drawn at levels denoted by Food and Nonfood indicate, respectively, the 87 and 66 percent increases these indices registered dur ing the 1970s. Although lines A and B both are consistent with the actual 87 percent increase in food prices that occurred during the 1970s, the differences in their slopes imply very distinct economic interpretations of this statistic. On one hand, lines B and C are compatible with the popular view that food prices increased at a relatively faster rate over this 10-year interval. That is, since 1970, the slope of line B, which represents a constant rate of growth for food prices, has been greater than the slope of line C, which depicts the growth rate of non food prices. This suggests that fundamental differences in production and marketing processes established different long-run growth rates for food and nonfood prices in the 1970s. Or, because the difference in slopes appears to be a perm anent structural difference, lines B and C also carry the implicit hypothesis that food will continue to increase in value, relative to nonfood products. Lines A and C also are consistent with the historical data but do not imply any fundamental changes in the relative growth rates of food and nonfood prices. In stead, line A illustrates the effect of certain events in 1973 on the relative level of food prices. But, aside from this isolated change caused by relative shifts in world food supply and demand relationships, lines A and C have the same slope. That is, with the exception of 1973’s adjustment in relative prices, both food and nonfood prices, on average, have grown at the same rate both before and since 1973. Therefore, lines A and C are consistent with the nominal price changes that occur during a neutral inflation. Or, stated differently, the slopes of lines A and C depict the shared increases in all nominal prices that are associated commonly with past rates of growth of the money stock. These theoretical relationships can be compared to plots of actual price changes shown in chart 1. In general, these plotted lines reflect the same qualitative results suggested by lines A and C in figure 1. The level of food prices did increase, relative to nonfood prices, in 1973 but, after the effects of this relative price change dissipated, food and nonfood prices tended to follow the same trend rate of growth. In fact, declines in the relative price of food in every year since 1978 have caused the food price and the nonfood price lines to converge. Or, rather, the large increase in the rela tive price of food during 1973-74 has been offset by five consecutive declines in relative food prices since 1978. APRIL 1983 Food Prices and Money Growth The distinctions of the two preceding sections sug gest that the problem for an analysis of food prices is to specify a statistical model that can distinguish between changes in relative and nominal prices or, alternative ly, betw een the types of change depicted by lines A and B in figure 1. One such model can be specified as: 4 (1) C P IF = a + X 1 1 S 1^ x M, _i + 2 d; x yt _ f + E a i= 0 j= 0 k= 0 RP,_ k + h X Z, + q X Z2 + e„ where CPIF is the CPI for food; M is the narrowly defined money stock, M l; y is real GNP; RP is the ratio of the Producer Price Indexes for the “food” and “non food” groups;4 Zj and Z2 are 0/1 dummy variables for phases I—II and phases III—IV, respectively, of Nixon administration price controls; b, d, g, h and q are estimated coefficients; t indicates time (quarterly in tervals, 1960-82); and et is a model error term. Dots over variable names indicate data m easured in growth rates. All data are seasonally adjusted. The reasoning behind this model of food price be havior derives from the basic considerations of figure 1 and the discussion of relative versus nominal prices.5 Because we know any observed change in food prices is likely to be apportioned in some m anner between changes in relative and nominal values, a model of price change must include variables associated with general inflation and with changes in product supplydemand relationships. Therefore, the model includes past growth rates of the money stock to account for that portion of changes in food prices that is associated with general inflation. Changes in the growth rate of real GNP are included to represent a cyclical effect on prices not captured by money growth. That is, if the equation of exchange is rew ritten as: P = M + V — v, then, for a given rate of increase in money and a given M l velocity, a higher rate of real income growth will tend to be associated with a slower rate of nominal price increase. Therefore, the signs on coefficients d. 'The actual commodity groups are the Producer Price Indexes for “all farm foods and feed and “all industrial com m odities,” respec tively; these groups represent, essentially, a “food” and “nonfood” division of the PPI. T h is same basic model, estim ated with monthly data, and a more detailed explanation of its theoretical support is found in Michael T. Belongia and Richard A. King, “A Monetary Analysis of Food Price D eterm ination,” American Journal o f Agricultural Eco nomics (February 1983), pp. 131-35. 7 FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Chort 1 Actual Movements in Food and Nonfood Prices 1970 1972 1974 1976 1978 a 19 80 1982 |J[ D a t a a r e from C o n s u m e r P ric e In d ic e s . are expected to be negative. Changes in basic food supplies are represented by a proxy of changes in the growth rate of relative food prices at the producer, or wholesale, level. The effects of official price controls from August 1971 through January 1974 are repre sented by variables Zj and Z2. Together, these vari ables encompass the sources and types of price changes discussed earlier. This model implies several specific hypotheses. First, a one-to-one relationship betw een past rates of money growth and nominal prices would be supported by a test of the full impact of all current and past values of M on CPIF; the specific hypothesis to be tested is: Digitized for 8FRASER 4 (2) 2 b, = 1, i= 0 or that an X percent increase in the rate of money growth over the most recent five quarters will cause a similar X percent change in the current growth rate of nominal food prices.6 6The postulated lag length is considerably shorter than th e 20quarter lag betw een money and prices reported in oth er studies. The reason for this difference is the choice of price index for the model’s dependent variable. Because supply and dem and func tions for food products tend to be more inelastic than those associ ated with other goods, changes in the supply of, or dem and for food will tend to affect prices more quickly than is typical in other markets. APRIL 1983 FEDERAL RESERVE BANK OF ST. LOUIS Another hypothesis concerns changes in relative prices. Here, the concern is the net impact of a change in the growth rate of real income and a change in relative producer prices. In addition to the effect of real activity on nominal price growth shown via the equation of exchange, a change in product supplies also could affect CPIF by changing the relative price of food. Because these effects are expected to be offset ting, the hypothesis test takes the form: 1 1 (3) 2 d, + 2 gk = 0. j= 0 k= 0 Finally, it is interesting to know w hether general price controls during the 1971-74 period had signifi cant effects on food prices, which were treated dif ferently than other controlled commodities. If controls were effective, the coefficient on Z x should be negative and the coefficient on Z2, when controls were gradually relaxed, should be positive. 0.58 0.097 0.66 M ,-i 0.345 2.25 M, —2 0.300 2.07 M.-3 0.155 1.02 M, 4 0.238 1.60 y. -0 .2 1 8 -1 .9 0 y,~, -0 .1 5 6 -1 .4 2 RP. 0.168 5.03 RP,~, 0.058 1.74 Z, -0 .8 3 8 - 2 .0 6 Zz 1.770 3.58 Hypothesis tests 4 F = 0.43 2 b, = 1 i= 0 1 2 1 dj + 2 II '’This relationship also appears to be stable over time. The model also was estim ated over 1960-72, 1970-82 and 1973-82 sub samples and, in each case, the growth rates of the money stock and food prices shared an approximate one-to-one correspondence. 0.165 M, gk = 0 F = 1.37 o 7Although the coefficients on the third and fourth lags of money growth are nonsignificant individually, an F-test on their joint significance suggests these term s should be retained in the model. t-statistics a II This result is supported by the tests of other a priori hypotheses. The net effect of changes in the growth rates of real income and relative producer prices is shown to be zero, indicating that relative food prices have not changed significantly over this sample period. This provides further support for the notion that food prices have increased, on average, in a fashion similar to general inflation. Therefore, as the discussion in the next section indicates, studies based only on factors affecting supply and demand conditions are in substan tial disagreement with the historical data: if relative prices have not changed appreciably, studies based on factors that shift supply and demand functions will not Coefficient estimates Variable o The ordinary least squares results in table 1 support these propositions. The hypothesis test for equation 2 suggests that the net impact of money growth is not significantly different from one; the rate of money growth over the current and past four quarters causes an equal change in the subsequent growth rate of retail food prices.7 Therefore, except for transitory short-run deviations, the observed changes in retail food prices have been changes in their nominal values, not in their relative prices. Changes in food prices are related most closely to changes in the growth rate of the money stock.8 Table 1 Estimated Results for Equation 1 Critical value for F1i79 = 3.97 (a = 0.05) R2 = 0.55 DW = 1.66 p re se n t accurate descriptions of observed price changes. Finally, the coefficients on price control variables are of the expected sign. From August 1971 through the end of 1972, when controls were applied most stringently, they apparently did reduce the rate of increase in reported food prices.9 Then, from 1973 through 1974, controls were relaxed gradually and food 'T his does not imply, however, that controls were an effective anti-inflationary policy. In fact, although there is an observed statistical effect on food prices in these results, controls themselves were abandoned, in large part, because of the resource allocation problems they caused. That is, controls masked changes in relative prices that give signals to producers concerning their output deci sions. Consider, for example, that higher food prices are caused by product shortages. H igher prices, however, will tend to encourage increased production and, in the longer run, increased production will cause lower prices. Therefore, if price controls limit or forbid price increases, their negative impact on production incentives will exacerbate the shortage-high price conditions. 9 FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 prices began to increase at a faster rate. These results again support expected price behavior during this period. The general conclusion of this analysis might be seen m ore clearly by constructing a comparison of the effects of M, y and RP on the growth rate of retail food prices. After adjusting C PIF for the effects of the mod el’s intercept, Zj and Z2, it is possible to write: _____ (1') C P IF ~ 4 _ 2 b; x M + i=0 1 2 dj x y + j=0 1 ___ 2 gk x R P k= 0 where the bars over variable names indicate their aver age, or mean, values. By summing the coefficient esti mates as indicated and inserting the data means, equa tion 1' can be rew ritten as: (4) 1.280 ~ (1.136 x 1.32) + ( - 0 . 3 7 4 x 0.77) + (0.226 x ( - 0 . 2 3 ) ) or, (5) 1.280 ~ 1.500 - 0 .2 8 8 - 0 .0 5 2 ~ 1.160. In this form, an evaluation of the model’s results at the data means indicates that M l and CPIF share an approxim ate one-to-one correspondence, whereas changes in real activity — over this sample period — tend to decrease the relative price of food. Contrary to the popular belief, food price increases would have been larger had it not been for the mitigating effects of real income growth and shifts in relative producer prices. NONMONETARY EXPLANATIONS FOR FOOD PRICE INCREASES: A CRITIQUE A num ber of studies have offered alternative ex planations for why food prices increase and, further, why they have increased relative to other prices. These explanations inclu d e increasing prices for farm products,10 farm price support program s,11 unioniza- 10See, for example, Don Paarlberg, Farm and Food Policy (Uni versity of Nebraska Press, 1980); Albert Eckstein and Dale Heien, “The 1973 Food Price Inflation,” American Journal o f Agricul tural Economics (May 1978), pp. 186-96; Rodney C. Kite and Joseph M. Roop, “Changing Agricultural Prices and Their Impact on Food Prices U nder Inflation,” American Journal o f Agricul tural Economics (D ecem ber 1981), pp. 956-61; and Lamm and W escott, “The Effects of Changing Input Costs . . . UJ. R. Penn, “Commodity Programs and Inflation, ”American Jour nal o f Agricultural Economics (D ecem ber 1979), pp. 889-95. Digitized for10 FRASER tion of food sector employees12 and increased concen tration of the food industry.13 The following discussion indicates that these explanations either are unrelated to the trend growth rate of food prices or predict results contrary to observed events. Rising Input Costs One alleged cause of increased food prices attributes observed increases in the CPIs for various food groups to increases in the prices of inputs used to produce finished retail food products. Specifically, some pre vious studies have found that increases in the nominal costs of raw farm products have led to subsequent increases in the retail prices of foods purchased by consumers. The logic behind this explanation is, essen tially, that if the prices of the inputs used to produce food items are increased, those processors and retailers who produce and sell food products also must raise their prices to maintain previous profit margins or avoid losses. The explanation that rising input costs have caused increases in retail food prices is flawed on an empirical basis, if for no other reason. That is, because the rela tive prices of major food groups at the producer level declined during most years of the 1970s, these inputs actually becam e relatively less expensive for food manufacturers. These declines in relative prices for raw farm products should have put downward pressure on both producers’ costs and output prices. Or, other things being equal, these data suggest that food manu facturers should have been able to produce a given quantity of food at lower — and declining — costs. This is an unlikely explanation for increasing retail food prices. Concentration Ratios and Prices Higher concentration ratios for the food industry or relatively higher union mem bership among workers in the food industry might explain why food prices are at a higher level than their values under perfect competi tion. But these structural characteristics of the indus try could only cause food prices to rise continuously if it is shown that these monopolistic elem ents also strengthened continuously over the same period. In stitutional arrangements — like union bargaining pow er and pricing strategies among a few relatively large 12R. McFall Lamm, “Unionism and Prices in the Food Retailing Industry,” Journal o f Labor Research (W inter 1982), pp. 69-79. 13Lamm, “Prices and C oncentration . . . .” FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 firms — usually act in a m anner similar to price support programs. That is, some degree of control over pricing decisions — such as a union’s ability to secure higher nominal wages for union workers — can act like a price support which raises a commodity’s price above its competitive market value. The ability of a union or a highly-concentrated food industry to raise wages or prices to higher levels, however, is not the same as an ability to raise relative wages or prices continuously. Again, there is a necessary distinction betw een rates of price change and changes in relative price levels. There are at least two reasons why neither type of m arket pow er is likely to explain ongoing price changes. On the one hand, a producer facing a down ward-sloping linear demand curve will have an incen tive to raise prices until profits are more affected by declining sales than by higher prices. If a firm starts at a position where raising prices is profitable and decides to raise its product’s price, the firm will benefit in two ways. The increased price will, ceteris paribus, reduce the quantity sold, which will reduce costs. At the same time, total revenue will increase because the per centage reduction in the quantity sold will be less than the percentage increase in the output price. At some point, where the product’s price elasticity is equal to — 1, total revenue will be maximized. At prices above this level, total costs will continue to decline but total revenue also will fall. Therefore, as Batten has ex plained, price increases beyond some level will result in reductions in marginal revenue (from a smaller quantity sold) larger than the associated decreases in marginal costs (from producing less).14 In this case, the price increases will reduce profits and, if other firms do not follow the price increases — as traditional oligopoly theory suggests — the firm’s market share also will be diminished. A second counterargum ent to the alleged rela tionship betw een increasing concentration ratios and inflation is found in the reason why an industry be comes more concentrated. Eckard, who found no relationship between concentration ratios and price increases, argues that industries become more concen trated because firms are able to produce at lower cost.15 The sequence of events begins with gains in productivity (most notably, labor productivity) that reduce a firm’s input costs and allow it to price its 14Dallas S. Batten, “The Cost-Push M vth,” this 1981), pp. 20-25. Review (June/July 15E. Woodrow Eekard, Jr., “Concentration Changes and Inflation: Some E vidence,” pp. 1044-51. Journal o f Political Economy (October 1981), output below the level charged by competitors. Conse quently, more efficient production and lower prices provide an opportunity for this firm to increase sales which, in turn, tends to make its industry more con centrated. This sequence of events — increased pro ductivity and lower input costs ultimately resulting in increased industry concentration — is supported by empirical evidence provided by Peltzm an.16 The con centration ratio-inflation hypothesis also suffers from its own predictions, however: if these models were correct, actual declines in the relative price of food must imply that the food industry has become less concentrated over this period. Union Power and Prices Similarly, the existence of union bargaining power might explain a higher level of costs for a firm purchas ing this type oflabor. And, a higher level of costs might be used to explain a higher price level for the products produced by a firm using union labor. For the same reasons used in the previous argum ent, however, the existence of bargaining power in wage negotiations is unlikely to explain why nominal or relative food prices would rise continuously. One extension of the sequence by which union pow er causes higher prices through increased wages is presented explicitly in a model by Moore and implicit ly in some food price studies.17 The argument pre sented is that union wage negotiations and their wage contracts are ongoing processes that result in con tinuous upward adjustments in nominal wage levels. Further, it is recognized that because wages are just one price among all prices, an increase in the relative price oflabor necessarily must be offset by a decline in the relative price of one or more other goods unless the money stock is increased. So, instead of an adjustment of relative prices and wages, the models argue that the Federal Reserve will monitor nominal wage increases and “ratify’’ them by increasing the money supply. Increases in the growth rate of the money stock will cause inflation, however, and therefore will reduce the purchasing power of wages as product prices increase. This reduction in purchasing power will, it is alleged, set off another round of wage increases to re-establish purchasing power. But, the effort is futile as the money ''’Sam Peltzman, “The Gains and Losses from Industrial C oncentra tion, Journal o f Law and Economics (October 1977), pp. 229-63. ''B asil J. Moore, “M onetary Factors,” in Alfred S. Eichner, ed., A Guide to Post-Keynesian Economics (M. E. Sharpe and Co., 1979), pp. 120-38'. 11 APRIL 1983 FEDERAL RESERVE BANK OF ST. LOUIS stock grows again and the rate of inflation increases further. Although a plausible explanation for ongoing in creases in food prices, this type of model rests on the assumptions that (a) wage increases established by union power cause increases in product prices, and (b) the Federal Reserve will ratify nominal wage increases with an expansion of the money stock. These are test able hypotheses of real-world behavior. But, an em pir ical investigation of these relationships rejected the notions that wage increases cause increases in food prices and that the growth rate of the money stock responds to changes in nominal wages.18 Therefore, in the one case when unions and food prices might be related, the statistical evidence does not support any direct linkage betw een wage rates and food prices. 1SM. Belongia, “A Note on th e Specification of Wage Rates in Cost-Push Models of Food Price D eterm ination,” Southern Jour nal o f Agricultural Economics (D ecem ber 1981), pp. 119-24. Digitized for 12 FRASER CONCLUSIONS Changes in food prices since 1970 have been attrib uted to a variety of sources. These explanations, how ever, often are based on some confusion over the basic distinction betw een isolated changes in relative prices and ongoing changes in nominal price levels. After accounting for this distinction, statistical analysis of the data suggest that the recent increases in food prices are increases in nominal price levels that share an approxi mate one-to-one relationship with past rates of money growth. Com peting explanations of food price be havior — unionization, oligopoly power and rising in put prices, among others — actually predict results that are contrary to the observed data over this period. Specifically, competing models are based on theories that predict increases in the relative price of food; in fact, the relative price of food has declined over much of the sample period. Relating money growth to food prices appears to offer a better explanation of what actually produced the food price increases during the 1970s, and what is likely to do the same in the 1980s. Polynomial Distributed Lags and the Estimation of the St. Louis Equation DALLAS S. BATTEN and DANIEL L. THORNTON c K J INCE its introduction in 1968 to investigate the relative impact of monetary and fiscal actions on eco nomic activity, the St. Louis equation has been the focus of considerable criticism .1 Much of this criticism stemmed from the fact that Andersen and Jordan’s conclusions were substantially different from those of the larger econometric models. In particular, they found that changes in the money stock have a sig nificant, lasting impact on nominal income, while changes in high-em ploym ent governm ent expendi tures and revenues, although having a short-run im pact, have no significant, lasting effect. Criticism of the St. Louis equation generally has fallen into two categories: the specification of the equa tion and the use of the polynomial distributed lag (PDL) estimation technique.2 The second category has The authors would like to thank R. Carter Hill and Thomas B. Fomby fo r their suggestions and comments. 'T he St. Louis equation first appeared in Leonall C. Andersen and Jerry L. Jordan, “M onetary and Fiscal Actions: A Test of Their Relative Im portance In Economic Stabilization,” this Review (November 1968), pp. 11-24. 2There have been three major criticisms of the specification of the St. Louis equation. First, since th e equation is not derived explicit ly from a structural macroeconomic model, relevant exogenous, right-hand-side variables may be excluded, and, as a result, the equation may be misspecified. See, for example, Franco Modi gliani and Albert Ando, “Impacts of Fiscal Actions on Aggregate Income and the Monetarist Controversy: Theory and Evidence,” in Jerom e L. Stein, ed., Monetarism, vol. 1, Studies in Monetary Economics (North-Holland, 1976), pp. 17-42; and Robert J. G or don, “Comments on Modigliani and A ndo,” in Monetarism, pp. 52-66. Second, failure to specify the appropriate indicators of monetary and fiscal actions may distort their exhibited relative importance. See Frank D e Leeuw and John Kalchbrenner, “M onetary and Fiscal Actions: A Test of Their Relative Im portance in Economic Stabilization — C om m ent,” this Review (April 1969), pp. 6-11; Edward M. Gramlich, “The Usefulness of Monetary and Fiscal Policy as Discretionary Stabilization Tools,” Journal o f Money, Credit, and Banking (May 1971), pp. 506-32; and E. Gerald C orri gan, “T he M easu rem e n t and Im p o rta n c e of Fiscal Policy C hanges,” Federal Reserve Bank of New York Monthly Review (June 1970), pp. 133-45. received far less attention in the literature, and inves tigations of it have been conducted in a far less sys tematic m anner than investigations of the other cate gory. Consequently, we have undertaken a thorough examination of the use of the PDL estimation tech nique to determ ine w hether the conclusions of the St. Louis equation are sensitive to either the lag structure employed or the polynomial restrictions imposed. A BRIEF SURVEY OF THE ST. LOUIS EQUATION The St. Louis equation has not changed substantially since its introduction. The original specification was: (1) AY, = a + + 3 2 3 ,A M t _, + 3 2 8 jARt j + e t, i= 0 3 2 i= 0 7, i= 0 where Y = nominal GNP, M = a monetary aggregate (either M l or the mone tary base), G = high-employment federal government expen ditures, Finally, ordinary least squares (OLS) estim ates of the param eters will exhibit simultaneous equation bias if the right-hand-side variables are not exogenous with respect to nominal income. See Stephen M. Goldfeld and Alan S. Blinder, “Some Implications of Endogenous Stabilization Policy,” Brookings Papers on Economic Activity (3: 1972), pp. 585-640; Robert J. Gordon, “Notes on Money, Income, and Gramlich, ” Journal o f Money, Credit, and Banking (May 1971), pp. 533-45; De Leeuw and Kalchbrenner, “Monetary and Fiscal Actions: C om m ent;” J. W. Elliott, “The Influence of Monetary and Fiscal Actions on Total Spending,” Journal o f Money, Credit, and Banking (May 1975), pp. 181-92; Keith M. Carlson and Scott E. Hein, “M onetary Aggregates as Monetary Indicators,” this Review (November 1980), pp. 12-21; and R. W. Hafer, “The Role of Fiscal Policy in the St. Louis Equation,” this Review (January 1982), pp. 17-22. 13 FEDERAL RESERVE BANK OF ST. LOUIS R = high-em ploym ent federal government rev enues and = error term .3 e The As indicate that all variables are first differences (i.e., AYt = Yt — Yt_ j). The coefficients of each lagged variable were constrained to lie on a fourth degree polynomial with both endpoint coefficients for each variable constrained to equal zero.4 In the original article, longer lag lengths were estimated but, since no coefficient past the third lag was statistically signifi cant, these lags were excluded. None of the reported results indicated any investigation of different lag lengths or different polynomial degrees for each vari able individually.0 In addition, equation 1 also was estimated in a modified form by combining the highemployment governm ent spending and revenue terms into the high-employment surplus/deficit (i.e., R-G). W hen Andersen and Carlson made the St. Louis equation the cornerstone of the St. Louis model, it contained the contemporaneous value and four lags of AM and AG; AR, however, was excluded from the equation.6 The same degree polynomial was em ployed, and the endpoint constraints were imposed. Many studies of the estimation of the St. Louis equa tion, both critical and supportive, appeared during the 1968-1975 period. These studies investigated, among other things, the sensitivity of the original results to the choice of lag structure and, indirectly, the ap propriateness of the restrictions imposed by the use of a PDL m odel.' Frequently, however, these studies 3Andersen and Jordan, “M onetary and Fiscal Actions. ” 4W ithout these constraints, the use of a PD L model would have been erroneous, as each variable in the original equation had only four coefficients in its lag structure while five param eters are needed to construct a fourth degree polynomial; the imposition of the endpoint constraints reduces the num ber of param eters to three. Thus, the use of a PD L model in the original St. Louis equation conserves three degrees of freedom. °A ndersen, in a subsequent paper, did investigate longer lag lengths (again with the same lag length specified for each variable) using the minimum standard error of the regression as the criterion for choosing the appropriate lag structure. He concluded that, based on the above criterion, the appropriate lag structure was longer than the one chosen originally, b u t that the qualitative results w ere not sensitive to the lag structure chosen. See Leonall C. Andersen, “An Evaluation of the Impacts of Monetary and Fiscal Policy on Economic Activity,” Proceedings o f the Business and Economic Statistics Section (American Statistical Association, 1969), pp. 233-40. 6Leonall C. A ndersen and Keith M. Carlson, “A M onetarist Model for Economic Stabilization,” this Review (April 1970), pp. 7-25. ‘P eter Schmidt and Roger N. W aud, “The Almon Lag Technique and the M onetary Versus Fiscal Policy D ebate,” Journal o f the American Statistical Association (March 1973), pp. 11-19; Elliott, “The Influence of M onetary and Fiscal Actions;” Leonall C. 14FRASER Digitized for APRIL 1983 made several changes simultaneously (e.g., employing different measures of monetary and/or fiscal policy actions and imposing a different polynomial degree and/or a different lag structure), so that it is difficult to identify the marginal impact of any individual change.8 Moreover, with one exception, the polynomial restric tions were never examined directly.9 Schmidt and W aud w ere the first to investigate the lag lengths for the individual variables of the St. Louis equation. They did so, however, within the framework of a fourth degree polynomial.10 They refrained from using endpoint constraints, arguing that the behavior of the polynomial outside of the range defined by the param eters is irrelevant. Using the minimum standard error as their criterion, they determ ined the appropri ate lag structure for the original equation to be six lags of AM, five lags of AG and seven lags of AR. Despite these changes, their results w ere not qualitatively different from those of Andersen and Jordan. Elliott attem pted to examine systematically the sen sitivity of the results to the choice of lag structure and the impact of the polynomial restrictions. Using a fourth degree PDL procedure, he estimated the equa tion as modified by Andersen and Carlson with four, eight and twelve lags for each variable. He also em ployed both ordinary least squares (OLS) and Shiller’s method of fitting lags with smoothness priors. His results indicated that the conclusions drawn from the estimation of the St. Louis equation do not depend importantly upon the lag structure chosen or the re strictions imposed by using a fourth degree PDL. Elliott did not conduct statistical tests of these proposi tions. Instead, he based his conclusions on a casual comparison of the results. Furtherm ore, he eonsid- Andersen, “An Evaluation of the Impacts of M onetary and Fiscal Policy on Economic Activity;” Corrigan, “The M easurem ent and Im portance of Fiscal Policy Changes;” D e Leeuw and Kalchbrenner, “Monetary and Fiscal Actions: C om m ent;” William L. Silber, “The St. Louis Equation: ‘Democratic’ and ‘Republican’ Versions and O ther E xperim ents,” The Review o f Economics and Statistics (Novem ber 1971), pp. 362-67; Gramlich, “T he Usefulness of Monetary and Fiscal Policy;’’ and Leonall C. A ndersen and Denis S. Karnosky, “The A ppropriate Tim e F ram e for C ontrolling Monetary Aggregates: The St. Louis E vidence,” in Controlling Monetary Aggregates II: The Implementation, Proceedings of a Conference Sponsored by the Federal Reserve Bank of Boston (Series No. 9, 1972), pp. 147-77. sFor example, see Corrigan, “The M easurem ent and Im portance of Fiscal Policy Changes;” Silber, “The St. Louis Equation: ‘D em o cratic’ and Republican’ Versions;” Gramlich, “The Usefulness of Monetary and Fiscal Policy;” and De Leeuw and Kalchbrenner, “Monetary and Fiscal Actions: C om m ent.” T h e one exception is Elliott, “The Influence of M onetary and Fiscal Actions.” 10Schmidt and W aud, “The Almon Lag Technique. ” FEDERAL RESERVE BANK OF ST. LOUIS ered only three possible lag structures (which were assumed to be the same for each distributed lag vari able) and only a fourth degree polynomial. After the Andersen-Carlson modifications of the original Andersen-Jordan equation, the only substan tive change in the equation took place as a result of an exchange betw een Friedm an and Carlson in the late 1970s.11 In updating the sample period over which the equation had been estimated, Friedm an noticed that the cumulative effect of government spending became statistically significant. In his response Carlson pointed out that when the original sample was ex panded, the standard error of the regression nearly doubled. This in d icated th at th ese errors w ere heteroscedastic.12 Using annual rates of change in place of the original first differences of the variables, Carlson respecified the equation.13 In this form, the errors were homoscedastic and the cumulative effect of government spending was no longer statistically sig nificant. Since the Friedman-Carlson exchange, the growth rate specification (or an approximately equiva lent alternative, first differences in natural logarithms) has been the widely accepted o ne.14 In summary, even though a num ber of studies have attem pted to investigate the effects of the lag length and PD L specification of the St. Louis equation, rel atively little work has been directed at investigating ''B enjam in M. Friedm an, “Even the St. Louis Model Now Be lieves in Fiscal Policy, "Journal o f Money, Credit, and Banking (May 1977), pp. 365-67; and Keith M. Carlson, “Does the St. Louis Equation Now Believe in Fiscal Policy?” this Review (February 1978), pp. 13-19. 12W hen the variance-covariance matrix is misspecified, the esti mated t-ratios are biased, and n either the direction nor extent of the bias can be determ ined a priori. See G. S. Watson, "Serial Correlation in Regression Analysis. I,” Biometrika (Decem ber 1955), pp. 327-41. I3This re-specification was proffered as an alternative to first differ ences in the original Andersen-Jordan article. John Vrooman, “Does the St. Louis Equation Even Believe in Itself? "Journal of Money, Credit, and Banking (F ebruary 1979), pp. 111-17, attempts to correct for heteroscedasticity in the first difference specification. He does so by dividing the observation matrix by the square-root of AYt. This transformation, however, creates correla tion betw een the error term and the right-hand-side variables — a violation of one of the classical assumptions of ordinary least squares estimation. 14See, for example, Keith M. Carlson, “Money, Inflation, and Eco nomic Growth: Some U pdated Reduced Form Results and Their Implications,” this Review (April 1980), pp. 13-19; Carlson and Hein, “Monetary Aggregates as M onetary Indicators;” John A. Tatom, “Energy Prices and Short-Run Economic Performance,” this Review (January 1981), pp. 3-17; Laurence H. M eyer and Chris Varvares, “A Comparison of the St. Louis Model and Two Variations: Predictive Performance and Policy Im plications,” this Review (D ecem ber 1981), pp. 13-25; and Hafer, “The Role of Fiscal Policy in the St. Louis E quation.” APRIL 1983 and testing the propriety of the polynomial constraints or the lag structure employed. Furtherm ore, most previous investigations have been conducted using the first difference specification of the equation. Thus, whether the policy conclusions drawn from the estima tion of the equation (especially for the growth rate specification) are influenced significantly by the choice of lag length and polynomial restrictions employed remains unresolved. POLYNOMIAL DISTRIBUTED LAGS The PD L estimation technique forces the coef ficients of each lagged variable of an equation to lie on a polynomial of degree p. In the presence of a high degree of multicollinearity, OLS estimates are not pre cise. Thus, the rationale for the use of the PD L tech nique is that it increases the precision of the estimates. Estimates of the individual lag weights, however, will be biased generally unless the correct lag length and degree of polynomial are specified.15 Therefore, it is important that the appropriate specification be deter mined. There are a num ber of procedures and criteria for determining the appropriate lag length and polynomial degree.16 W e use a computationally efficient proce dure outlined recently by Pagano and Hartley (here after PH ).17 Details of the PH technique and other relevant considerations are presented in the appendix. W hen Almon first introduced PD L models, she sug gested that endpoint constraints always be employed. 15Let £, p and £*, p* denote th e assumed and correct lag length and degree of polynomial, respectively. Estimates of the param eter vector will be biased if (a) £ = £* and p < p*, (b) 6 < 8 * and p = p* or (c) £ > £*, p = p* and £ — £* > p*. In the instance w here £ — £* =£ p*, the polynomial distributed lag estimates may be biased, but need not be. That is, there are restrictions that may or may not be satisfied by the data. Furtherm ore, PD L estimators will be inefficient if £ = £ * a n d p > p * . See P. K. T rivediandA . R. Pagan, “Polynomial D istributed Lags: A Unified Treatm ent, Economic Studies Quarterly (April 1979), pp. 37-49. 16See Trivedi and Pagan, “Polynomial D istributed Lags: A Unified Treatm ent;” D. F. H endry and A. R. Pagan, “D istributed Lags: A Survey of Some R ecent D evelopm ents,” unpublished manu script; Robert J. Shiller, “A D istributed Lag Estim ator D erived from Smoothness Priors,” Econometrica (July 1973), pp. 775-88; J. D. Sargan, “The Consum er Price Equation in the Post War British Economy: An Exercise in Equation Specification Testing,” The Review o f Economic Studies (January 1980), pp. 113-35; and George G. Judge and others, The Theory and Practice o f Econ ometrics (John Wiley and Sons, Inc., 1980), chap. 11. 17See Marcello Pagano and Michael J. H artley, “O n Fitting D istri buted Lag Models Subject to Polynomial Restrictions, "Journal o f Econometrics (June 1981), pp. 171-98. 15 FEDERAL RESERVE BANK OF ST. LOUIS The suggested endpoint constraints take the form P b+ i = 3 - i = 0, where £ is the chosen lag length. Although the end point constraints put explicit restrictions on the dis tributed lag weights outside of their relevant range, they also imply homogeneous restrictions on the lag weights inside the range via homogeneous restrictions on the polynomial coefficients.18 Thus, the endpoint constraints add two additional homogeneous restric tions for each PD L variable to those already implied by the PDL model. The problem is that endpoint con straints have no basis in either economic or econo metric theory, as Schmidt and W aud have pointed o u t.19 As a result, they represent a set of ad hoc restric tions whose sole purpose is to increase the efficiency of estimation. Nevertheless, their validity can be tested. APPLICATION TO THE ST. LOUIS EQUATION To investigate the appropriate lag lengths and polynomial degrees for the St. Louis equation, we employ the growth rate specification20 J Yt = a + 2 i= 0 K M ,- i + 2 i=0 -yi G ,_ j + et. The dots over each variable represent quarter-toquarter annualized rates of change, and Y, M and G represent nominal GNP, money (the M l definition) and high-employment governm ent expenditures, re spectively. The estimation period considered is II/ 1962 to III/1982. Lag Length Selection The first step of the PH technique is to select lsThis can be seen by noting that the endpoint constraints require 5o + S,( —1) + S , ( - l )2 + . . . + S,,( —l)1’ = 0 and 8,, + 8,(6+ 1) + 5,(S!+ l)2 + . . . + SpU+1)1’ = 0. These restrictions can be w ritten as R8 = 0, because for a PDL model, = H 8 , so that = H +J5, w here H + = Therefore, R8 = R H +Ji = R*f} = 0. Thus, the endpoint con straints impose a set of homogeneous restrictions R* on p. See Daniel L. Thornton and Dallas S. Batten, “E ndpoint Constraints and the St. Louis Equation: A Clarification,” Federal Reserve Bank of St. Louis Research Paper No. 83-001 (1983). 19See Schmidt and W aud, “The Almon Lag T echnique,” p. 12. 20W e chose to employ this specification because it is the one in cluded in the St. Louis model. For a com plete specification of the St. Louis model, see the appendix to Keith M. Carlson, “A Mone tary Analysis of the Administration’s Budget and Economic Pro jections,” this Review (May 1982), pp. 3-14. Digitized for16 FRASER APRIL 1983 appropriate lag lengths 0 , K) for money and govern m ent expenditure growth. Once these lag lengths are selected, a re-application of the technique results in the selection of the polynomial degrees.21 The PH procedure is somewhat complicated when appropriate lag lengths and polynomial degrees must be selected for two variables.22 The use of the PH technique, like other procedures for specifying a distributed lag model, requires the choice of a maximum lag length (L). W e considered two choices of L: 12 and 16.23 An application of the PH technqiue to the St. Louis equation results in a choice of 10 lags on M and 9 on G. This selection is basically consistent with the results of a standard F-test.24 Ordinary least squares estimates of this lag specification, as well as the usual specification with four lags on both M and G, are presented in table 1. Note that the standard error of the regression is reduced substantially and the adjusted R2 is increased substantially by including the additional distributed lag variables. Furtherm ore, the coefficients on the longest lag terms are significant in the longer lag spec ification. These results suggest that this specification is preferable. Indeed, a likelihood ratio test of the restric tions implied by the current specification rejects them at the 5 percent level.20 Nevertheless, it is interesting to note that the con clusions about the long-run efficacy of monetary and fiscal policy are unaffected by the choice of lag struc ture. The hypothesis of the long-run ineffectiveness of money can be rejected for both lag specifications; the 21Standard statistical procedures cannot be used to select the lag length if the polynomial degree is specified first. See footnote 6 of the appendix for further details. 22The choice of lag length and polynomial degree also involves sequential hypothesis testing. As we note in th e appendix, care m ust be taken in conducting sequential tests. Given the problems with sequential tests (and those of prelim inary test estimation), we initially chose a relatively low significance level of 15 percent, opting to guard against incorrectly excluding relevant components of the distributed lag. As a general rule, one would have expected the chosen lag length to be shorter had we used a m ore common significance level, such as 5 percent. In our case, the lag specifica tion would have been th e same had we selected a 5 percent significance level. 23The results for L = 16 w ere identical to those for L = 12. Thus, the PH technique seems to b e relatively insensitive to the choice of L. 24W ith L = 12 for both \1 and G, the_F-statistic calculated to test the hypothesis that the 10th lag on M is significant was 2.45*. The F-statistic calculated for the same test for th e 8 th and 9th lags on G w ere 2.55* and 1.77, respectively. (The * indicates significance at the 1 0 percent level.) 2SThe likelihood ratio statistic was 32.13, which compares with a critical value of x 2 ( H ) of 19.68 at th e 5 percent level. FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Table 1 Ordinary Least Squares Estimates of Alternative Lag Length Specifications of the St. Louis Equation, 11/1962-111/1982 Estimated Coefficients Variable Constant PH Specification Current Specification 2.342 (1.56) 1.643 (107) M0 M, m2 m3 m4 m5 m6 m7 m8 m9 M10 SM 0.767* 0.635* 0.295 -0 .3 7 7 * 0.233 -0 .1 2 7 -0 .1 3 4 -0 .1 2 6 0.297 0.230 -0 .5 3 0 * 1.163* (4.61) (3.66) (1.80) (2.36) (1.38) (0.68) (0.79) (0.74) (1.69) (1.15) (2.77) (4.50) 0.474* 0.441* 0.356* -0 .1 7 9 0.022 (3.37) (3.09) (2.51) (1.22) (0.15) Go G, g2 g3 g4 g5 g6 g7 g8 g9 ZG 0.110* 0.056 -0 .0 9 5 * 0.028 -0.001 -0 .0 4 2 0.095 0.047 -0 .1 1 6 * -0 .1 1 6 * -0 .0 3 4 (2.34) (1.24) (2.11) (0.61) (0.03) (0.90) (1.93) (0.92) (2.32) (2.33) (0.26) 0.108* 0.034 -0 .0 9 6 * 0.040 -0 .0 0 4 (2.21) (0.71) (2.04) (0.84) (0.09) 0.082 (0.82) 1.114* (4.69) SE = 3.21 SE = 3.58 R2 = 0.47 R2 = 0.33 DW = 2.17 DW = 2.01 ‘ Indicates significance at the 5 percent level. Absolute value of t-statistics in parentheses. same hypothesis about government expenditures can not be rejected. Polynomial Degree Selection The chosen lag structure is used in the selection of the appropriate polynomial degree. The appropriate polynomial degree is selected by re-parameterizing the model and applying the same technique used to select the lag length. A direct application of the PH technique to the question of polynomial degree selection results in selecting a ninth degree polynomial on M and a seventh degree polynomial on G. The results of con ventional F-tests, however, indicate that there are more restrictive specifications that cannot be rejected at the 5 percent level. Given that the polynomial re strictions tends to smooth out the distributed lag weights and, thus, might result in more accurate out-of-sample forecasts, we decided to present the results of both the PDL specification resulting from a strict application of the PH technique and the one determ ined by em ploying the greatest num ber of polynomial constraints that satisfy a conventional F-test at the 5 percent level. The latter specification has a sixth degree polynomial on M and a third degree polynomial on G. The results of the estimation of these specifications (denoted A and B, respectively) and the PD L specification presently used (denoted C) are given in table 2. These equations were estimated with restricted least squares (BLS).26 We believe RLS is preferable to the standard PDL method because it makes the param eter restrictions explicit and permits ease in testing the individual and joint PDL restrictions. It is clear from these results that each of the two longer lag PD L specifications performs better than the current one. Each has a smaller standard error and a larger adjusted R2. Nevertheless, it is interesting to note that the tests of the long-run efficacy of the mone tary and fiscal policy variables also are insensitive to the PDL specification. The long-run effect of money is not significantly different from one, while the long-run effect of government expenditures is not significantly different from zero, for all three specifications.27 The short-run distributed lag response patterns, however, differ significantly. Tests of the Endpoint Constraints As we noted earlier, endpoint constraints represent ad hoc restrictions and, thus, should not be employed routinely. Nevertheless, since the current specifica tion of the St. Louis equation employs polynomial restrictions only in the form of endpoint constraints, we decided to test these constraints for all three spec ifications. The results of these tests for the relevant joint and individual restrictions are presented in table 26For a discussion of th e equivalence betw een standard PD L estimation and RLS, see Judge and others, The Theory and Prac tice o f Econometrics, pp. 640-42. 2‘Estimates of two other PD L specifications yielded the same con clusions regarding the efficacy of m onetary and fiscal policy. See the appendix for details of these specifications. 17 APRIL 1983 FEDERAL RESERVE BANK OF ST. LOUIS Table 2 Estimates of Various PDL Specifications of the St. Louis Equation, 11/1962-111/1982______________________________ Estimated Coefficients Variable Constant A B C 2.366 (1.56) 2.608 (1.63) 1.799 (1.16) (4.14) (5.01) (1.56) (2.15) (0.57) (0.63) (1.85) (0.51) (1.27) (2.30) (3.50) (4.38) 0.557* 0.677* 0.198* -0 .0 5 3 -0.06 1 -0 .0 3 7 -0.08 1 -0 .0 8 7 0.114 0.355* -0 .5 0 1 * 1.081* (3.90) (5.01) (2.27) (0.57) (0.78) (0.42) (1.05) (0.96) (1.20) (2.19) (2.64) (3.96) 0.461* 0.458* 0.244* 0.015 -0 .0 9 2 (3.87) (5.62) (2.46) (0.19) (0.76) Me m7 m8 m9 M10 2M 0.642* 0.771* 0.236 -0 .3 1 2 * 0.075 0.080 -0 .2 4 3 -0 .0 8 0 0.209 0.410* -0 .6 4 5 * 1.143* G0 G, g2 g3 g4 g5 g6 g7 g8 g9 2G 0.118* 0.039 -0 .0 6 8 -0 .0 0 2 0.011 -0 .0 1 6 0.041 0.096* -0 .1 2 5 * -0 .1 2 0 * -0 .0 2 6 (2.52) (0.88) (1.64) (0.06) (0.31) (0.43) (1.10) (2.18) (2.54) (2.42) (0.19) 0.106* 0.022 -0 .0 1 6 -0.02 1 -0 .0 0 8 0.012 0.024 0.016 -0 .0 2 7 -0 .1 1 6 * -0 .0 0 8 (2.32) (0.80) (0.58) (0.82) (0.35) (0.54) (0.94) (0.60) (1.07) (2.53) (0.07) 0.094* 0.022 -0.04 1 -0 .0 2 6 0.034 (2.18) (0.65) (1.12) (0.77) (0.78) 0.110 (0.82) M0 M, m2 m3 m4 m5 1.086* (4.52) SE = 3.24 SE = 3.42 SE = 3.65 R2 = 0.46 R2 = 0.39 R2 = 0.31 DW = 2.27 DW = 2.41 DW = 2.17 'Indicates significance at the 5 percent level. Absolute value of t-statistics in parentheses. Specification A has ninth degree and seventh degree polynomials on M and G, respectively. Specification B has sixth and third degree polynomials on M and G, respectively. Specification C is the current specification with four lags on both M and G and endpoint constraints. 3. The test of all four endpoint constraints rejects these constraints for both specifications A and B, but not for the current specification. The head constraint on M, however, is never rejected by the F-test, and the tail constraint is rejected only for specification B. Never theless, in general, the endpoint constraints do not fare well when applied to the longer lag specifications. Out-of-Sample Forecast Comparisons While it is clear that the alternative PD L repre sentations of the St. Louis equation perform better on an in-sample comparison, it is interesting to see how Digitized for18 FRASER well they perform on the basis of out-of-sample fore casts. To this end, we estim ated these specifications from 11/1962 to a term inal period and forecasted out-ofsample for four quarters. W e then added four quarters to our estimation period, re-estim ated the equation and repeated the process. W e did this for six periods beginning with a terminal date of III/1976, generating 24 out-of-sample forecasts of the growth of nominal GNP. The root mean square errors (RMSEs) of these forecasts are summarized in table 4. Both the PH specification and the current specification do about equally well by a RMSE criterion over the entire period; there are significant differences, however, in APRIL 1983 FEDERAL RESERVE BANK OF ST. LOUIS Chart l F o re c a s t E rro rs o f A lt e r n a t iv e S p e c ific a t io n s o f the St. L o u is E q u a t io n Actual-Predicted Percent Actual-Predicted P er cent Table 3 Tests of Endpoint Constraints for Various PDL Specifications of the St. _____________ Louis Equation F-Statistics for Constraints Specification/ Variable Tail Head Head and tail Specification A M 3.22 1.99 1.61 G 3.66 8.42* 4.21* 3.15* M and G Specification B M 2.40 7.09* 3.59* G 6.46* 6.86* 4.72* 1976 Specification C M 0.81 1.84 1.13 G 1.83 4.11* 2.18 1.68 M and G ’ Indicates significance at the 5 percent level. Table 4 Root Mean Square Error of the Forecast for Various Specifications of the St. Louis Equation ______________ Period IV/1976-111/1982 A B C 4.77 4.49 4.70 IV/1976-111/1977 4.13 2.77 2.98 IV/1977-111/1978 3.42 5.31 6.28 IV/1978-111/1979 5.35 3.81 2.02 IV/1979-111/1980 4.17 2.89 4.17 IV/1980-111/1981 6.29 5.96 4.87 IV/1981-111/1982 4.72 5.16 6.25 1977 1978 1979 1980 1981 1 982 3.74* M and G their subperiod forecast performances.28 The most re stricted PDL specification shows an improvement over the current specification, reducing the out-of-sample RMSE by nearly 5 percent over the period and produc ing a smaller RMSE of the forecast in four of the six sub periods. A graph of the out-of-sample forecast errors for specifications B and C is presented in chart 1. It is clear from chart 1 that both specifications produce similar patterns of forecast errors over the period. The only significant exception occurs in the third quarter of 1982, when specification B underpredicts nominal GNP growth by about as much as specification C over predicts it. SUMMARY AND CONCLUSIONS This paper has investigated the lag length and polynomial degree specifications of the St. Louis equa 2S()ne could argue that the result may be biased in favor of our PDL specification because the lag structure was chosen over the entire period. Indeed, the lag structure appears to lengthen during the latter part of the sample. The estim ated lag structure for the period ending III/1976 was four on M and six on C. Thus, the lag structure chosen was nearly that of the current specification. The PDL specification was a first degree polynomial on M and a sixth degree on G. W hen this specification was used to forecast out-ofsample, it performed somewhat worse than the current specifica tion, with a RMSE of 4.89. O ur estimates indicate that the lag structure lengthened when the terminal date of the sample period was extended to III/1979. If the shorter lag structure were used over the first three subperiods and the longer lag structure (spec ification B) used over the last three, the RMSE for the entire period would be 4.39, somewhat better than either specification alone. 19 FEDERAL RESERVE BANK OF ST. LOUIS tion to determ ine w hether its conclusions about the long-run efficacy of monetary policy and inefficacy of fiscal policy are affected by the lag length employed or its polynomial distributed lag specification. In so doing, we have employed a computationally efficient method for determ ining the appropriate lag length and polynomial degree of a general polynomial distributed lag model. Our results indicate that the important policy con clusions of the St. Louis equation are insensitive to the lag length specified and to the polynomial restrictions imposed. In particular, the long-run effectiveness of money growth and the long-run ineffectiveness of growth in high-employment government expenditures are substantiated by ordinary least squares estimates of model param eters using both the Pagano-Hartley- APRIL 1983 determ ined lag length and the current lag length specifications, as well as by estimates of several PDL specifications. Thus, there is no evidence that the con clusion of the St. Louis equation can be traced to these types of econometric misspecification. We did find a PDL specification that outperforms the current specification by both in-sample and out-ofsample criteria. This specification has considerably longer lags on both the monetary and expenditure variables and more polynomial restrictions. Finally, we found that the Pagano-Hartley tech nique, used in conjunction with standard F-tests, is a convenient and com putationally efficient tool for selecting the lag length and polynomial degree of a PDL model. APPENDIX Pagano and Hartley have recently developed a methodology for determ ining the appropriate lag length and degree of polynomial which is computa tionally efficient.1 In order to illustrate the use of the Pagano-Hartley (PH) technique, consider the general distributed lag model K (A.l) s* Y, = 2 |xkZkt + 2 pjXt_j + et, t = l , 2, ..., T, k= 1 j=0 where et ~ NID (0, cr2), and where Z^, is the kt!l independent variable and Xt is an independent vari able which affects Yt with a lag of length £*. The polynomial distributed lag (PDL) model in volves imposing restrictions on the (3 coefficients such that Pj = 80 + 5J + S^j2 + . . . + 8P. jp*. That is, each of the individual lag weights falls on a polynomial of degree p*, where p* < £*.2 These re strictions can be written more compactly in matrix notation as J3 = H8, a (£* + 1) by (p* + 1) matrix of coefficients.3 Substitut ing the above restrictions into the model, we get K (A.l') Y, = p* 2 ^ kZkt + 2 S„X*, k=l q=0 where X*t = £ (Xt^ jh j + 1, q+1) and where hj + 1, q+1 j=0 is the (j + l)th, (q+ l)th elem ent of H, j = 0, 1, 2, ... C* and q = 0, 1, 2 ,..., p*. It is clear that imposing the polynomial restrictions reduces the num ber of param eters by £ * - p * and, thus, imposes £* —p* homoge neous restrictions on the param eter vector j}. Thus, estimating equation A .l' is tantamount to estimating equation A .l subject to homogeneous restrictions of the form RJJ = 0, where R is a (£* —p*) by (£* + 1) matrix.4 It should be apparent that the validity of the Specifically, H takes the general form 4 0 l 8 0 l 2 >’* e *2 e *3 C*P* 1 1 1 0 1 2 0 1 e* i whereJ3 = (p0 Pi • • . Pj*)',8_= (8081,..8p*)', and H is 'Pagano and Hartley, “On Fitting D istributed Lag M odels.” 2Strictly speaking, p* could equal C*; however, there would be no polynomial restrictions. Thus, it is doubtful that one would de scribe a model as a PD L if p* = 5*. Digitized for20 FRASER '‘T here are a num ber of ways of generating the restriction matrix, R. See Shiller, “A D istributed Lag Estim ator;” and Judge and others, The Theory and Practice o f Econometrics (John W iley and Sons, Inc., 1980), pp. 642-44. FEDERAL RESERVE BANK OF ST. LOUIS polynomial restrictions, including the endpoint con straints, can be tested easily.5 Of course, the correct values of the lag length and degree of the polynomial are generally unknown. Since the selection of an im proper lag length or polyno mial degree generally leads to biased coefficient esti mates, the selection of £ and p is extremely important. The selection process, however, is not easy. For one thing, the appropriate lag length cannot be deter mined using standard procedures if the degree of the polynomial has been selected.6 Even though a num ber of techniques have been suggested for selecting £ and p, the PH method was chosen, in part for its computa tional convenience.7 The PH method proceeds by determ ining the lag length and then the degree of the polynomial. The PH technique can best be illustrated by rewriting equation A .l in matrix form as (A.2) Y = Zji + Xp + e, where Z and X are T by K and T by (£* + 1) matrices of observations on the independent variables, and y. and J3 are K by 1 and (£* -I-1) by 1 vectors of parameters. The procedure begins by choosing a maximum lag length L. Equation A.2 with the maximum lag length can be rew ritten as (A.3) Yl = WL 4»l + e„ where W L = [Z:XL], and]J>L = [jo.: 0 L]'. The observa tion matrix W L is then decomposed to APRIL 1983 WL = q ,n l by the Gram-Schmidt decomposition. H ere Q l is a matrix whose columns form an orthonormal basis for the column space of W L, and NL is an upper triangular matrix with positive diagonal elem ents.8 Equation A.3 now can be rew ritten as Y l = Q l A l + £l > where XL = [ V \ x £ ] ' = N l ^ (, Given that QL is orthonormal, the least squares esti mate of_XL is given by h . = [> :A lT = Q l'X l , and the structural param eters can be obtained from NL 4»L =_Al- An advantage of the PH m ethod comes in noting that the elements of Al are mutually independent random variables. In particular, xf xf ~ N ID (Xh a 2), i = 0 , 1 , 2 , ..., e* ~ N ID (0 , <r2), i = c* + l , e* + 2 , . .., L. Pagano and Hartley note that there is a one-to-one correspondence betw een the null hypothesis involving the 0s and the Xs. Given this and the orthogonality of the PH procedure, the following sets ofhypotheses are equivalent: H l - j: 3 l = Pl -1 = ••• = Pl - j = 0 j = 0, 1, 2, ..., L T h e re are a num ber of alternative norms for testing these restric tions. See Judge and others, The Theory and Practice o f Econo metrics, p. 646. T h is is seen by noting that, once the polynomial degree is selected, alternative lag specifications am ount to imposing the polynomial restrictions on different param eter spaces. Thus, restrictions on the lag length are non-nested when p is specified. See Peter Schmidt, “A Modification of the Almon D istributed Lag "Journal o f the American Statistical Association (Septem ber 1974), pp. 67981; and H endry and Pagan, “D istributed Lags: A Survey of Some Recent D evelopm ents.” In this regard, it would be appropriate to use the maximum R2 criterion as Schmidt and W aud do; however, this procedure may lack power. A more useful procedure has been suggested by Pesaran. N either procedure, however, provides in formation concerning the degree of polynomial. See Schmidt and W aud, “The Almon Lag Technique”; and M. H. Pesaran, “On the G eneral Problem of Model Selection,” Review o f Economic Studies (April 1974), pp. 153-71. O ne attractive method has been suggested by H endry and Pagan, “D istributed Lags: A Survey of Some Recent D evelopments. ” This procedure involves a sequence of hypothesis tests commencing with an initial arbitrary choice of a lag length. W hile this procedure has potential merit, it is not w ithout its difficulties. Furtherm ore, it may involve an extremely laborious test procedure w hen th ere are two PD L variables, as in the St. Louis equation. For another procedure, see Sargan, “The C onsum er Price Equation in the Post W ar British Econom y.” H£_ j:X£ = X?._, = ... = x t . j = 0 j = 0, 1, 2, . .., L. Hence, the Gram-Schmidt decomposition provides a convenient basis for testing the null hypothesis that there exists a lag length, £, such that the null hypoth esis P p = 0 can be rejected. If no such £ can be found, then there is no distributed lag of X. The test of the simple hypothesis X^-j = 0 can be carried out by a t-test of the form t L- j = X ^ -j/s j = 0, 1, 2, ..., L, where - Y| - Q[ Al . T h e Gram-Schmidt procedure is often used w hen the observation matrix is ill-conditioned. If th e diagonal elem ents are chosen to be positive, as they are in our case, Q Land NL are unique; see G. A. F. Seber, Linear Regression Analysis (John W iley and Sons, Inc., 1977), chapter 11. 21 FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Because of their common divisor, these t-statistics are not independent; however, they are uncorrelated.9 Pagano and Hartley also suggest that the above hypotheses are equivalent to H ' L - j:XE_ j = 0 j = 0, 1, ..., L , due to the orthogonality of their procedure. These hypotheses, however, are not equivalent in any direct sense. To see this, recall that A l = where NL is an upper-triangular matrix with positive diagonal elements. The ith row of NL can be repre sented as N ’l = (0, ..., 0, Tlii, Tlii + 1, ..., TliL), where is the ith-jth elem ent of NL. Thus, the hypothesis test that = 0 is given by Mi = tIlPl = 0. Likewise, the test that M,-i = = 0 is given by + tilP l = 0, and so on. Thus, the hypotheses of H 'L-j are really tests of linear combinations of the distributed lag weights, where the particular linear combination is determ ined by the elem ents of rows of NL. In practice we found that the absolute value of the diagonal ele ments of N l tended to be somewhat large relative to the off-diagonal elem ents for the lag length selection and very small relative to the off-diagonal elem ents in the polynomial selection. In the former case, there fore, testing the hypothesis that = 0 was very near testing the hypothesis that P, = 0, while in the later case it was closer to the null hypothesis H* j. Given this, we decided to supplem ent the use of t-tests on the Xs with conventional F-tests of the equivalent hypotheses of H and H*. W e recommend that one investigate the NL matrix to identify the na ture of the hypotheses being tested when using the PH t-statistics. We should note also that the use of the PH method is complicated somewhat by the presence of two distrib uted lag variables on the right-hand side. One can readily see that, in view of the upper-triangular form of N l, hypothesis tests involving a second distributed lag will not be consistent with H* -, unless the GramSchmidt procedure is applied to each set of distributed lag regressors separately. Unfortunately, the resulting sets of jointly orthogonal regressors will not them selves be orthogonal to each other. As an alternative, we ran two separate Gram-Schmidt regressions with each distributed lag variable entered last. F urther more, we did this by reducing by one the lag length or polynomial degree for one variable and holding the maximum lag length or polynomial degree for the other variable (which was entered last) constant. In this way, we determ ined w hether the lag length chosen for one variable was affected by the lag length specified for the other. Of course, we were particularly concerned that the lag length selected for one be the same if the chosen lag length of the other was used instead of L. The procedure had the added advantage of allowing us to calculate an L by L matrix of F-statistics for all possible combinations of lag structures (or in the case of PDL selection, degrees of polynomials) from L ortho gonal regressions.10 Hypothesis Testing Considerations W hen determ ining the “correct” lag length using either the t-tests or the F-test, care must be taken in choosing a critical value on which to test the null hypothesis. Two considerations are important. First, the null hypotheses Ht_j: \£_j = 0 j = 0, 1, 2, ..., L represent a set of sequential hypotheses. It is usually assumed that these hypotheses are nested so that if any one is true, the preceding hypotheses must be true also and, if any one is false, so must be the succeeding ones. Thus, the null hypothesis becomes more restricted as each successive test is conducted, and the probability of committing a Type I error increases. If we let denote the significance level of the jth test, it can be shown that the probability of committing a Type I error for the jth test, a,, is a ■= 1 i 5, \ £ j(l-a j-i) + a j- i ifj = 1 ifj 3= 2. Thus, the probability of rejecting the null hypothesis when it is true will rise as the length of the lag is reduced. Anderson suggested that one would like to balance the desirability of not overestimating the lag length with the sensitivity to non-zero coefficients.11 He recommends setting L fairly large, but letting £j be 10This can be seen by noting that the RSS w hen j lags are om itted is given by RSSi K L -j-1 k=l k= 0 = Y, ’Y, - S (\£)2 - 2 „ (\£)2. 1'Anderson also provides a test procedure for orthogonal regressors 'T his perm its the use of t-tables from Seber. See Seber, Regression Analysis, pp. 404—5. Digitized for22 FRASER Linear which have some optimal properties; however, the test is som e what cumbersome. SeeT . W. Anderson, The Statistical Analysis o f Time Series (John Wiley and Sons, Inc., 1971), pp. 30-43. FEDERAL RESERVE BANK OF ST. LOUIS small for j near L. While no optimal rules exist, Ander son suggests (A.4) {j = g(L + 1 — j), j = 1, 2, 3 ........... L L for subsequent tests. An alternative would be to use the t-tables from Seber. In addition to the above problem, we have the prob lem that an estimator based on a prior test is a prelimi nary test estimator. While nothing is known about such estimators when the sequence of tests is greater than one, it is known that, in the case of one pre-test, the estimator has a risk function which may exceed that of O LS.12 Furtherm ore, the difference betw een the risk of the preliminary test estimator and OLS increases as the significance level is reduced. While the optimal critical value will vary with the particular choice of loss function, the evidence suggests that standard signifi cance levels of 5 or 10 percent may be below the optimal level for one pre-test.13 These considerations, coupled with the fact that overestimates of the lag length are less likely to result in bias than underesti mates, suggest that one may want to consider an initial value of the significance level that is fairly large.14 POLYNOMIAL DEGREE SELECTION Having selected a lag length, £, the next step is to determ ine a polynomial degree, p. This can be accom12The risk function is E[(ip* —q>)'X'X(ip* —9 )], w here <p* is the p re test estim ator of 9 . l !For example, Sawa and Hiromatsu have shown that the standard critical values of the t-statistic arc substantially above the optimal critical values in the case of a mini-max regret loss function with one restriction. On the other hand, Toyoda and Wallace have shown that OLS should always be chosen when the num ber of linearly independent restrictions are less than five if one wishes to minimize the average regret. See Takamitsu Sawa and Takeshi Hiromatsu, “Minimax Regret Significance Points for a Prelim i nary Test in Regression Analysis,” Econometrica (November 1973), pp. 1093-1101; andT . T oyodaandT. D. Wallace, “Optimal Critical Values for Pre-Testing in Regression,” Econometrica (March 1976), pp. 365-75. 14To guard against incorrectly excluding com ponents of the distrib uted lag or imposing invalid polynomial restrictions, an initial significance level of 15 percent was chosen. The critical t-values for testing each successive hypothesis are as follows: j t-value 1 2 3 4 5 6 7 8 9 10 11 12 1.46 1.51 1.56 1.61 1.67 1.74 1.81 1.90 2.00 2.12 2.30 2.57 APRIL 1983 plished by simply re-applying all of the procedures outlined above to the PDL model with lag length £. To see this, write the model with the selected lag length as (A. 5) Y e = Z^l + + Fj,. Recall that = H8 where H is (£ + 1) by (p* + 1) and 8 is (p* + 1) by 1. Thus, this equation can be rew ritten as (A.6) Y f — Zp. 4- X CH8 + f^ (A.6') Y* = Z\x = X*8 + e*. It is clear from this expression that the choice of a polynomial degree p is completely analogous to the choice of the lag length above, where the maximum degree of the polynomial considered, p, initially is set equal to £.15 EMPIRICAL RESULTS In applying the PH technique, we initially chose a maximum lag length of 12; however, we also consid ered L = 16. The PH t-statistics for those runs with both M and G last are given in table A .I. This proce dure chose 10 lags on M and 9 on G for L = 12 and 16. We then chose these lags for one variable and let the other be set at L = 12. The results were unchanged. These results also appear in table A .I. Furtherm ore, F-tests of the restrictions implied by this section were basically consistent with the PH results, when L was set at 12 (see footnote 24 of the text). This was not true, however, for L = 16. In this instance, the presence of a num ber of insignificant coefficients prior to the first significant one diluted the calculated F-statistic so that a very short lag would have been chosen by an F-test. Thus, the PH t-statistics appear to be less sensitive to the choice of L than the standard F-test. Letting the maximum degree polynomial be 10 for M and 9 for G, we then re-applied the PH technique to loPagano and H artley offer an equivalent two-step procedure, which is not discussed here. See Pagano and Hartley, “On Fitting D istributed Lag Models Subject to Polynomial Restrictions.” As an efficient alternative to eith er of these approaches, one could employ the stochastic information from the lag length selection process with the nonstochastic information in the design matrix in a Theil-Goldberger mixed estimation procedure similar to Schil ler’s Bayesian method. Fomby has shown that such stochastic restrictions can be tested under a generalized mean square error norm. See H. Theil and A. S. G oldberger, “On Pure and Mixed Statistical Estimation in Economics,” International Economic Re view (January 1961), pp. 65-78; Thomas B. Fomby, “MSE Evalua tion of Shiller’s Smoothness Priors,” International Economic Re view (February 1979), pp. 203-15; and Judge and others, The Theory and Practice o f Econometrics, pp. 652-53. 23 APRIL 1983 FEDERAL RESERVE BANK OF ST. LOUIS Table A.1 Pagano-Hartley t-statistics for Lag Length Selection G with t o n M equal to M with 8 on G equal to 16 12 9 16 12 10 0 4.84 5.45 5.42 2.68 2.67 2.72 1 4.49 4.33 4.61 1.04 1.13 1.16 2 2.51 2.36 2.24 -1 .8 4 -1 .8 9 -1 .9 0 3 -2 .2 0 - 1 .7 3 -1 .7 1 0.97 0.96 1.01 4 0.28 0.09 0.60 0.23 0.17 0.19 -1 .2 2 Lag 5 -1 .9 6 -2 .0 5 -2 .1 1 -0 .8 9 -1 .2 1 6 -0 .4 2 -0 .0 1 -0 .4 7 1.34 1.37 1.41 7 -0 .4 2 -0 .6 1 -0 .4 3 0.58 0.44 0.44 8 0.77 0.88 1.22 -2 .3 0 -2 .3 8 -2 .3 4 9 -0 .5 0 0.10 -0 .1 3 -2 .2 2 * -2 .2 2 * -2 .3 2 * 10 -2 .5 8 * - 2 .7 0 ' - 2 .7 2 ' -0 .3 0 -0 .5 8 -0 .6 5 11 0.09 -0 .1 3 0.19 0.93 1.18 1.20 12 -0 .1 0 0.17 0.31 0.98 0.64 0.68 13 -0 .5 7 14 0.41 1.01 15 -0 .8 2 - 1 .2 4 16 0.19 1.28 1.15 'First significant t-statistic Table A.2 Pagano-Hartley t-statistics for Polynomial Degree Selection Polynomial Degree M with p on G equal to 9 G with p on M equal to 10 0 3.44 - 0 .2 7 1 -5 .7 3 -2 .8 5 2 2.84 -0 .1 7 3 -2 .1 7 -2 .7 7 4 -2 .3 4 0.73 5 -0 .4 8 1.05 6 -2 .3 2 1.12 7 0.44 2.55 8 1.11 -1 .6 5 * 9 -1 .8 5 * -0 .4 7 10 1.26 'First significant t-statistic. determ ine the polynomial degree. The PH t-statistics are presented in table A.2. The PH technique selected a ninth degree polynomial on money and an eighth degree polynomial on government expenditures for the same significance level as used before. W hen we re-estimated the equation on the lower degree polyno Digitized for 24 FRASER mials, however, the coefficient of the eighth degree on G failed to be significant. The seventh was significant, regardless of the lag length on M. Thus, the PH tech nique suggests a ninth degree polynomial on M and a seventh degree on C. This implies only one polynomial restriction on M and two on G. (An F-test of these restrictions could not reject the null hypothesis. The calculated F-statistic was 1.43.) Furtherm ore, the matrix of F-statistics of all possible polynomial restrictions on a PD L model with 10 lags on Kl and 9 on G, given in table A.3, suggests that even more restricted models could pass an F-test. Clearly, a num ber of different polynomial degree specifications satisfy an F-test at the 5 percent level. W e can see, for example, that had we chosen the polynomial degree on M first and then selected the polynomial degree on G, we would have chosen a fourth degree polynomial on M and an eighth degree polynomial on G. Alternatively, had we investigated G first, we would have chosen a seventh degree polynomial on G and a sixth on M. These are circled in table A. 3. W e could also choose the polynomial degree by selecting the most restricted model that passes an F-test at, say, the 5 percent level. This criterion would select a sixth de gree polynomial on M and a third degree on G. This F-statistic is bracketed in table A.3. All four of these FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Table A.3 F-statistics for Testing Polynomial Restrictions on M and G Degrees for G Degrees for M 0 0 1 2 3 4 5 6 7 8 9 4.09 4.13 4.38 4.53 4.75 5.08 5.47 5.62 5.76 6.32 3.37 1 3.00 2.64 2.80 2.82 2.92 3.10 3.32 3.20 3.05 2 2.78 2.46 2.61 2.58 2.65 2.79 2.99 2.87 2.50 2.79 2.51 3 2.80 2.46 2.63 2.54 2.57 2.64 2.82 2.68 2.24 4 2.49 2.13 2.30 2.10 2.21 2.26 2.43 2.13 (EzD 2.02 5 2.61 2.26 2.45 2.28 2.41 2.49 2.69 2.40 2.02 2.37 6 2.58 2.17 2.37 [1 .9 6 ] 2.10 2.21 2.37 CT46) 1.33 1.62 7 2.77 2.35 2.59 2.14 2.33 2.51 2.77 1.74 1.63 2.09 8 3.02 2.56 2.84 2.27 2.54 2.83 3.20 1.82 1.75 2.52 1.60 — 9 3.03 2.48 2.79 2.09 2.37 2.63 2.94 1.43 0.87 10 3.13 2.51 2.86 2.06 2.37 2.69 3.16 1.48 0.22 PDL specifications — the one selected by the PH technique and the three indicated in table A. 3 — were estim ated; however, only the results for the one selected by the PH technique and the most restricted specification are presented in this paper. The results of the other specifications were similar to those of the most restricted PD L specification and, hence, are not reported h e re .16 16The hypothesis tests concerning the effects of monetary and fiscal policy yielded conclusions identical to those reported here. The out-of-sample RMSEs of the forecast for the period III/19Y6—III/ 1982 w ere smaller than the RMSEs of specifications A or C. 25 Weekly Money Supply Forecasts: Effects of the October 1979 Change in Monetary Control Procedures R. W. HAFER JL HE activity of most financial market participants on Friday afternoons can be predicted with great accuracy: they anxiously will be awaiting the 4:15 p.m. EST announcem ent of the new weekly money stock data. Despite the fact that the weekly data are con taminated by a great deal of “noise, ” a fact that greatly reduces the data’s usefulness in revealing any policy trend, market participants still wager large sums and reputations on correctly anticipating the elusive week ly money figure.1 The impact of unanticipated changes in the weekly money supply on short-term interest rates has been investigated extensively. In general, the evidence shows a positive relationship betw een unanticipated changes in money and movements in market rates.2 Although this empirical relationship existed through1See David A. Pierce, “T rend and Noise in the M onetary Aggre gates,” in Federal Reserve Staff Study, New Monetary Control Procedures, vol. II (February 1981), especially pp. 19-22. Pierce estimates that the noise in weekly money data is around $3 billion dollars, assuming an aggregate level of $400 billion. As he notes, “In general, these results are further evidence that very little can be inferred from any bu t the most atypical movements in weekly data” (p. 2 2 ). out the 1970s, the relative impact of weekly money “surprises” on short-term interest rates has been great er since the October 1979 change in monetary control procedures. In fact, over 25 percent of the volatility of the 3-month Treasury bill rate during the time period of the money supply announcem ent can be attributed directly to the increased volatility of unanticipated weekly changes in m oney since O ctober 1979.3 Moreover, unanticipated money supply changes that lie outside the Federal Reserve’s announced money growth range appear to have a relatively greater effect on interest rates than money surprises falling within the announced growth range.4 The evidence clearly indicates that unanticipated changes in the money stock have an important effect on interest rates. Consequently, examining the character istics of the money supply forecasts that give rise to such behavior is important. Several studies have ex amined the weekly money supply forecasts for the period prior to October 1979; but little has been done on com paring the forecasts across the announced change in monetary control procedures.5 The purpose of this article is to analyze the effects of the October 2 See, for example, Jacob Grossman, “The ‘Rationality’ of Money Supply Expectations and the Short-Run Response of Interest Rates to M onetary Surprises,” Journal o f Money, Credit and Banking (November 1981), pp. 409-24; V. Vance Roley, “The Response of Short-Term Interest Rates to Weekly Money A nnouncem ents,” W orking Paper No. 82-06, Federal Reserve Rank of Kansas City (Septem ber 1982); Thomas Urich, “The Information C ontent of Weekly Money Supply A nnouncem ents,” Journal o f Monetary Economics (July 1982), pp. 73-88; and Thomas J. Urich and Paul W achtel, “ M arket R esponse to th e W eekly M oney Supply Announcements in the 1970s,” Journal o f Finance (Decem ber 1981), pp. 1063-72. For another interpretation, see Bradford Cor nell, “Money Supply A nnouncements and Interest Rates: Another View "Journal o f Business (January 1983), pp. 1-23. Digitized for26 FRASER 3 Roley, “The Response of Short-Term In terest Rates.” 4 Ibid. See also, Neil G. Berkman, “On the Significance of Weekly Changes in M l,” 1978), p p . 5-22. New England Economic Review (M ay-June 5Studies investigating the forecasts prior to the O ctober 1979 policy shift are Grossman, "The Rationality’ of Money Supply Expecta tions,” and Thomas Urich and Paul W achtel, “The Structure of Expectations of the Weekly Money Supply A nnouncem ent,” (New York University, February 1982; processed). Roley, “The Re sponse of Short-Term Interest R ates,” provides some evidence on this issue for the period February 1980 to N ovem ber 1981. FEDERAL RESERVE BANK OF ST. LOUIS 1979 change in monetary control on the weekly money supply forecasts. U nder the assumption of rational ex pectations, a change from one recognized monetary control procedure to another should have no effect on the forecast characteristics.6 In other words, a change from one m onetary control procedure to another should not affect the unbiased and efficiency aspects of the forecasts. If, however, the new procedure is not “well-defined” — that is, the rules of the game are changing constantly — then weekly money supply forecasts may appear biased and inefficient.7 WHAT DOES “RATIONALITY” IMPLY? The theory of rational expectations is based on the prem ise that market participants construct forecasts of the future in a m anner that fully reflects the relevant information available to them. Because wealth-maxi mizing individuals will not make forecasts that are continually wrong in the same direction, the rational expectations approach suggests that forecasts of eco nomic phenomena should be unbiased. Moreover, if the forecast errors could not have been reduced by using other available information, then forecasters have efficiently utilized the relevant data at their dis posal. The issue investigated here is w hether the weekly forecasts of the M l money stock change have been affected noticeably by the October 1979 change in monetary control procedures. More specifically, the question asked is: assuming rational expectations, has ®The concept of rational expectations is based on the belief that economic agents are utility maximizers. Thus, market participants form expectations that fully reflect all available information. More formally, rational expectations imply that individuals’ subjective probability distribution of possible outcomes is identical to the objective probability distributions that actually occur. Conse quently, the only way policymakers can affect behavior is to “fool” the people in an inconsistent manner. This concept is developed more fully in John F. M uth, “Rational Expectations and the Theory of Price M ovem ents,” Econometrica (July 1961), pp. 315-35; Robert E. Lucas, Jr. “Expectations and the Neutrality of M oney,” Journal o f Economic Theory (April 1972), pp. 103-24; Robert J. Rarro, “Rational Expectations and the Role of Monetary Policy,” Journal o f Monetary Economics (January 1976), pp. 1-32; and Thomas J. Sargent and Neil Wallace, “Rational Expectations, the Optimal M onetary Instrum ent, and the Optim al Money Supply Rule, "Journal o f Political Economy (April 1975), pp. 241-54. im p lic it in this is the presum ption that market participants will expend resources to decipher the new policy procedures and adapt their forecast formation process accordingly. This does not seem unreasonable given the sophistication of financial market analysts in gauging actual Federal Reserve behavior. For a discussion on the transition from one policy to another and the implications for rational expectations, see Benjamin M. Friedm an, “Optimal Ex pectations and the Extrem e Information Assumptions of Rational Expectations’ M acrom odels,” Journal o f Monetary Economics (January 1979), pp. 23-41. APRIL 1983 the change in monetary control procedures affected the unbiased and efficiency characteristics of the week ly money supply forecasts? If the forecasts from the post-October 1979 period are not different than those from before, we then would conclude that the fore casters have adapted to the new policy regime. If they differ, however, the evidence would not reject the hypothesis that they have been unable to ascertain the policymaker’s behavioral rule.8 Three sample periods are used in the following anal ysis. The full period is from the week ending January 11, 1978, to the week ending June 16, 1982. Given the change in operating procedures in late 1979, the rele vant subperiods are from the week ending January 11, 1978, to the week ending October 3, 1979, and from the week ending October 10, 1979, to the week ending June 16, 1982.9 W ith these sample periods, the un biased and efficiency characteristics of the weekly money supply forecasts across the change in monetary control procedures can be investigated. Weekly Money Supply Data The money data series used in this article are the actual and expected, initially announced week-to- sThe dilemma facing market participants is known as the “Lucas problem .” Essentially, even though individuals act rationally in making their forecasts — that is, use all of the information thought to be relevant — failure to account for a procedural shift will lead to incorrect forecasts. Thus, forecasting guidelines used under one procedure may not apply under another. For the specific problem tested here, it may be the case that the announced policy differs from that actually followed. If policy actions are not characterized easily, that is, if policy is unpredictable, then forecasts may be biased and inefficient simply because agents have not determ ined the structure of the model. For a discussion of this concept, see Robert E. Lucas, Jr., “Econom etric Policy Evaluation: A C ri tique,” in Karl B runner and Allan H. M eltzer, eds., The Phillips Curve and Labor Markets, The Carnegie-Rochester Conference Series on Public Policy (vol. 1, 1976), pp. 19-46. Bradford Cornell recently has argued that apparent irrational behavior on the part of market participants evidenced by biased and inefficient forecasts, may very well be due to the change from a predictable policy regim e to one that continues to be unpredict able. As he states, “O n O ctober 6 [1979], market participants suddenly discovered that even the rules of the game w ere subject to change. As a result, they began studying weekly money supply figures not only with the goal of determ ining what the current policy was, but also with the goal of determ ining how the rules of the game might be changed.” In this sense, market participants face a perpetual “Lucas problem .” See Cornell, “Money Supply Announcements and In terest Rates: A nother View,” p. 21. 9Note that the post-O ctober 1979 period includes the period of credit controls, essentially the second quarter of 1980. This period is included because an examination of the error pattern from week ly money forecasts indicated no difference betw een this period and any other. Moreover, market participants continued to forecast weekly money changes throughout the control period. 27 FEDERAL RESERVE BANK OF ST. LOUIS week changes in the narrowly defined money stock (Ml). Figures for the actual changes in M l are taken from the Federal Reserve’s H .6 weekly statistical re lease. Because the sample covers a period of changing definitions, the following guideline is used: From January 11, 1978, to January 31, 1980, the weekly money supply changes are based on the old definition of M l. From February 8, 1980, to November 20, 1981, the money stock is defined as the actual M1B measure, not the M1B figure that was adjusted for NOW account movements. Finally, from November 27,1981, to June 16, 1982, the data are based on the then-current defini tion of M l. The data used as a measure of the market’s forecasts were obtained from Money Market Services, Inc.10 Since 1977 this firm has conducted a weekly telephone survey of 50 to 60 governm ent securities dealers to get their expectations of the impending change in money. Prior to early 1980, the poll was conducted twice a week, on Tuesdays and Thursdays. Since then, howev er, only the Thursday survey has been conducted con sistently, because of the shift in the Federal Reserve’s announcement of the weekly money supply figures from Thursday to Friday afternoon. For our purposes, therefore, we employ the mean of the Thursday survey responses.11 Are Weekly Money Forecasts UnbiasedP Forecasts of weekly changes in the money stock are unbiased predictors of the actual change if the actual and forecasted values differ only by som e random term. Mathematically, this requirem ent can be stated (1) AM, = t_j AM,E + e, where AM, is the actual change in the money stock, ,_ xAMf is the expectation held in period t-1 for the change in the money stock in period t, and e, is a random error term with zero mean and variance of. 10It has been argued that survey data are not good measures of the market’s expectations of some macroeconomic variable. This argu m ent is founded on the belief that most survey respondents are not actual market participants. In other words, th eir responses to the survey are not based on some profit-maximizing behavior that has generated the forecast. The weekly money forecasts used here are taken from dealers actively participating in th e financial mar ket, thus reducing the force of this criticism. See Edward J. Kane and Burton G. Malkiel, “Autoregressive and Nonautoregressive Elem ents in Cross-Section Forecasts of Inflation, ” Econometrica (January 1976), pp. 1-16. 11For an analysis of the Tuesday and Thursday forecasts, see Gross man, “The ‘Rationality’ of Money Supply Expectations.” This analysis covers only the period 1977 to 1979. Digitized for 28 FRASER APRIL 1983 To test for the absence of bias, equation 1 is rew rit ten and estimated as (2) A M , = a 0 + (3, t - i A M f + e, where a 0 and (3] are the param eters to be estim ated.12 In this form, the weekly money forecasts are unbiased predictors of actual money supply changes if the joint hypothesis that a 0 = 0 and Pi = 1 cannot be rejected. Moreover, the estim ated residuals from this regression (et) should not exhibit serial correlation if the forecasts are unbiased predictions of the actual change in money. Table 1 presents the regression results from estimat ing equation 2 using the expected and actual money stock changes. The full-period results suggest that the forecasts of weekly changes in the money stock are unbiased predictors of the actual changes. The calcu lated F-statistic does not exceed the critical value of 3.04 at the 5 percent significance level. Consequently, the null joint hypothesis that a 0 = 0 and = 1 is not rejected. Moreover, the residuals of the equation show no indication of first-order serial correlation, as evi denced by the Durbin-W atson statistic. Thus, the weekly money supply forecasts appear to be unbiased across the full sample. To see if the forecasts are unbiased before and after the October 1979 change in monetary control proce dures, equation 2 was re-estim ated for the two periods January 11, 1978, to October 3, 1979, and October 10, 1979, to June 16, 1982. These regression results also are reported in table l . 13 The estimates from the pre-O ctober 1979 period again indicate that the forecasts are unbiased. The calculated F-statistic is not statistically significant, and the Durbin-W atson statistic again indicates no firstorder serial correlation among the residuals. In con trast, the post-October 1979 regression results perm it us to reject the hypothesis that the forecasts are un biased predictors of the actual changes. Although the estimated constant term is statistically insignificant, the hypothesis that the estim ated slope term (pi) does not differ from unity is rejected easily (t = 2.33). Consequently, the joint hypothesis underlying this 12This type of test is used widely in studies of expectations data. F or studies examining money stock forecasts, see, for example, Gross man, “The ‘Rationality’ of Money Supply Expectations;’’ Urich and W achtell, “The Structure of Expectations;” and Roley, “The Response of Short-Term In terest Rates.” 13This dichotomization of the sample is supported statistically by Chow-test results: the calculated F-value is F(2,228) = 3.93, which exceeds the critical 5 percent level. FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Table 1 Test Results for Bias Equation Estimated: AMt = a0 + t- Estimated coefficients1 Period «0 Summary statistics2 0. R2 DW F3 1/11/78-6/16/82 -0 .0 4 4 (0.30) 1.207 (10.43) 0.32 1.87 1.65 1/11/78-10/3/79 -0 .3 5 2 (1.65) 1.060 (6.54) 0.32 1.89 1.60 10/10/79-6/16/82 0.181 (0.91) 1.373 (8.60) 0.35 1.86 3.70 'Absolute value of t-statistics appear in parentheses. 2R2 is the adjusted coefficient of determination; DW represents the Durbin-Watson test statistic. The reported F-statistic is used to test the null hypothesis that ( a 0 , (3,) = (0,1). 3The relevant 5 percent critical F-values are: January 11, 1978, to June 16, 1982 — 3.04; January 11, 1978, to October 3, 1979 — 3.10; and October 10, 1979, to June 16, 1982 — 3.07. test also is rejected; the calculated F-statistic of 3.70 exceeds the 5 percent critical value of 3.07. Thus, the evidence suggests that forecasts of weekly money supply changes have been biased since the October 1979 change in implem enting monetary policy. Are Weekly Money Forecasts Efficient? The efficiency condition requires that forecasts fully reflect all pertinent and readily available informa tion.14 Since the information available to individuals includes the past history of the series being forecast, it is possible to test the hypothesis that the forecasts are “weakly” efficient; that is, at least the information con tained in the history of weekly money supply changes is used efficiently. This concept of efficiency requires that the process actually generating observed changes in weekly money and the process generating the fore casts of these changes are the same. The simplest process to assume is an autoregressive one, where observed and expected changes are generated solely by the past history of the series itself. Mathematically, this concept of efficiency can be stated as (3) n AM, = 2 Pi A M ,_ i + m ,, i= l 14O f course, additional information will be acquired only if the marginal benefits are at least as large as the marginal costs of acquisition. A useful discussion of this point is provided in Armen A. Alchian, “Information Costs, Pricing, and Resource U nem ploym ent,” in E dm und S. Phelps, and others, Microeconomic Foundations o f Employment and Inflation Theory (W. W. Norton & Company, Inc., 1970), pp. 27-52. n (4) ,_ ,A M tE = 2 Pi' A M ,_ | + (i2t, i= l where (Xk and (x2t are random error terms. In this format, weak-form efficiency requires that (3j = (3- for all i; i = 1, 2 ,..., n .15 To determ ine if survey respondents efficiently uti lized the information contained in past weekly money supply changes, equation 4 is subtracted from equation 3, yielding the estim ated equation n (5) AM, - ,-iA M f = b„ + 2 b* AM, , + <(>„ i = l where the dependent variable AM, — , _ iAM,K repre sents the forecasters’ errors in predicting weekly m oney changes, and th e in d ep e n d e n t variables, AMt _j, are the actual changes in m oney.16 The equa tion permits a constant term (b0) to be estimated in stead of subsuming it into the error structure, which is represented by the term <J>t (= |X|, — |x2t). The null hypothesis to be tested is that the estim ated bi (= (3j — 15This form of the efficiency test was proposed in James E . Pesando, “A Note on the Rationality of the Livingston Price Expectations Data "Journal o f Political Economy (August 1975), pp. 849-58. 16The lagged values of data used in the efficiency test are the one-week revised num bers, not the initially reported weekly figures. Since the revised figures contain more information than the originally released data — th e data contained in the revision itself — using original data would deprive forecasters of some information. It should be noted, however, that th e conclusions reached were not affected when originally reported data was used to generate lagged changes in the money stock. 29 FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 Table 2 Test Results for Weak-Form Efficiency 4 Equation Estimated: AMt - t -iA M tE = b0 + 2 bjAMt _j + 4>t i= 1 Estimated coefficients1 Period bo b2 b, Summary statistics2 ^3 b4 R2 DW F3 1/11/78-6/16/82 0.139 (0.87) -0 .0 4 2 (0.69) -0 .0 6 7 (1.14) 0.087 (1.48) 0.005 (0.09) 0.01 1.92 0.81 1/11/78-10/3/79 -0 .2 5 9 (1.24) 0.026 (0.28) -0 .0 7 7 (0.84) -0.021 (0.23) -0 .0 1 7 (0.20) 0.01 1.95 0.28 0.398 (1.77) -0.07 1 (0.90) -0 .0 6 8 (0.90) -0 .1 1 2 (1.48) 0.007 (0.10) 0.02 1.93 0.82 10/10/79-6/16/82 'See notes accompanying table 1. 2See notes accompanying table 1. The reported F-statistic is used to test the null hypothesis that b, (i = 1,2,3,4) = 0. 3The relevant 5 percent critical F-values are: January 11,1978, to June 16, 1982 — 2.41; January 11, 1978, to October 3, 1979 — 2.48; and October 10, 1979, to June 16, 1982 — 2.44. P/) are not statistically different from zero for all i (i = 1, 2,..., n) as a group. Moreover, the estimated error structure should not exhibit serial correlation.17 Table 2 presents the results of estimating equation 5 for the period January 11, 1978, to June 16, 1982. Four lags were chosen to capture the informational content of past changes in weekly money. The regression re sults indicate that past changes in the money supply do not explain any significant portion of the forecast error. The calculated F-statistic (0.81) is far below acceptable critical values. The Durbin-W atson statistic also indi cates that serial correlation is not present among the residuals. Thus, for the full period, we cannot reject the hypothesis that forecasters efficiently used the in formation contained in past changes in the money stock in forming their predictions. We next test the efficiency hypothesis for the preand post-October 1979 periods; these empirical results also are found in table 2. In both instances, we again cannot reject the hypothesis that past information about weekly money changes was used efficiently. Neither F-statistic is significant at the 5 percent level. Based on these results, therefore, the weak-form effi ciency hypothesis is not rejected by the data, regard less of the sample used. 17See Donald J. Mullineaux, “On Testing for Rationality: Another Look at the Livingston Price Expectations D ata , ”Journal o f Polit ical Economy (April 1978), pp. 329-36 for a discussion of this test. 30 Tests of Stronger-Form Efficiency The above evidence suggests that forecasts of weekly money stock changes are weakly efficient. Efficiency, however, also may be considered in a broader sense. This broader efficiency criterion requires that forecasts incorporate all of the relevant and available informa tion. Thus, similar to the previous hypothesis, efficien cy in the broad sense requires that the forecast errors be orthogonal, or systematically unrelated to all rele vant available information sets.18 To test this concept of efficiency, we estimate the equation n (6) AM, - t_| AMf = c0 + 2 C; It_j + w„ i= 0 where I,_j refers to lagged values (i = 0, 1,..., n) of information that are not incorporated in past money stock changes, and wt is another random error term. The analysis is intended to determ ine w hether the survey resp o n d en t’s weekly errors in forecasting money supply changes can be explained by some set(s) of information that are readily available. If the esti- 18Tests using this stronger form of efficiency are presented in Gross man, “The ‘Rationality’ of Money Supply Expectations,” and, using interest rate expectations data, in Renjamin M. Friedm an, “Survey Evidence on th e ‘Rationality’ of Interest Rate Expecta tions, "Journal o f Monetary Economics (October 1980), pp. 45365, w here the phrase “information orthogonality” was coined. FEDERAL RESERVE BANK OF ST. LOUIS APRIL 1983 t- 1 AMf = c0 + 1 + S Equation Estimated: AMt - II M3 O 0 Table 3 Test Results for Stronger-Form Efficiency Calculated F-statistics Period Information set 1/11/78-6/16/82 1/11/78-10/3/79 10/10/79-6/16/82 Consumer and industrial loans 3.551 1.74 2.941 Demand deposits at large weekly reporting banks 4.251 0.51 3.571 Float 0.60 0.55 0.61 Adjusted base 3.291 0.55 3.651 1Significant at the 5 percent level of confidence. The relevant critical F-values are: January 11,1978, to June 16,1982 — 2.26; January 11, 1978, to October 3, 1979 — 2.32; and October 10, 1979, to June 16, 1982 — 2.28. mated e, coefficients are not significantly different from zero as a group, then we cannot reject the strongerform hypothesis of efficiency. If contrary evidence is found, then the results would suggest that forecasters could have reduced their prediction errors by using the information sets investigated here. It is, of course, impossible to account for every imag inable information set that each forecaster could have used. Consequently, we analyze several sets of in formation that are available on a timely basis and are potentially useful in estimating future money stock developments. The information sets used are consum er and industrial loans, demand deposits at large week ly reporting banks, float and the adjusted monetary base as defined by the Federal Reserve Bank of St. Louis. In all cases, the data used are taken from origi nal Federal Reserve statistical releases that were avail able to forecasters prior to the weekly money stock announcem ents.19 Although we realize that the series 19A11 data are in term s of level changes from the previous week. Data sources are the Federal Reserve H4.1 and H4.2 statistical re leases, and the Federal Reserve Bank of St. Louis. This procedure may im part some m easurem ent error since only the initially released data are used. Given the short tim e horizon used and the observation that th e weekly data revisions are not severe, the approach used seems sufficient. It also should be noted that, since February 1980, data on consum er and industrial loans and dem and deposits at weekly reporting banks have been released concurrently with the money supply num bers. Thus, these two series offer no prior information during the post chosen do not exhaust the set of possible information sources, they are sufficiently broad to test the hypoth esis at hand. Table 3 reports the calculated F-statistics from estimating equation 6 using the different information sets. In each test, the information set contains contem poraneous and four lagged terms. The outcome for the full period suggests that forecasters efficiently utilized the information contained in the float information set: the reported F-statistic is not large enough to reject the null hypothesis. The results for the other information sets — consumer and industrial loans, demand de posits at large weekly reporting banks and the adjusted base — reject the efficiency hypothesis. For these, the F-statistics exceed the 5 percent critical value (2.26), implying that forecast errors could have been lessened if the information contained in these data had been used. Equation 6 was re-estimated for the pre- and postOctober 1979 periods; these results also are found in table 3. The full-period results are dominated by the post-October 1979 period. Prior to the shift in control procedures, forecasters’ predictions of weekly money supply changes appear to have efficiently incorporated the information sets tested here: all the F-statistics are less than the 5 percent critical value (2.32). In contrast, February 1980 period. They do, however, provide more informa tion that forecasters may use in generating their expected money numbers. 31 the post-October 1979 results reveal that, except for float, the forecasters could have improved upon their ability to predict changes in the money stock by incor porating the information contained in the series on loans, demand deposits and the adjusted base. Thus, over the recent period, the forecasts do not m eet the broader efficiency criterion tested here. CONCLUSION Previous examinations of survey data on weekly money supply forecasts have focused primarily on the effects of unanticipated money changes on market in terest rates. Although several studies have examined the forecasts’ rationality, there has been no systematic investigation into the effect of the change in monetary control procedures on the unbiased and efficiency characteristics of the forecasts. 32 The evidence presented here indicates that the change in control procedures has had a significant effect on the characteristics of weekly money supply forecasts. Prior to October 1979, the forecasts of the change in the weekly money stock were unbiased and efficient. In contrast, weekly money forecasts since October 1979 have been biased and inefficient. The results of this investigation lend support to the recently suggested hypothesis that, since O ctober 1979, “market participants [have] concluded that the rules under which monetary policy is conducted could no longer be considered constant. ”20 If this indeed is true, then the combined evidence from this study and those dealing with the interest rate effects of unantici pated money supply changes suggests that a more predictable control procedure would contribute to a more stable financial market. ^C ornell, “Money Supply A nnouncem ents,” p. 22.