Seasonal Dummies Research Articles

Intertemporal Basis Behaviour on the Chicago Wheat Futures Market

The study consists of an economic analysis of the behaviour of the Chicago wheat basis during the two decades from 1960 to 1980. First, there is a descriptive analysis of changes to the basis between the 1960's and 1970's. Second, an econometric model which can be used to explain observed and possible future movements in the basis is estimated.The econometric model provides for the direct estimation of both intra- and inter-seasonal movements of the basis. In developing this model a single-equation “quasi reduced-form” model incorporating the theoretically important factors determining the supply and demand for storage and the resulting price from month to month is specified.Model estimation involves the incorporation of a lagged dependent variable and a set of seasonal dummy variables to account for the cyclical behaviour of the basis. The Generalized Least Squares regression results reveal that the basis lagged one period and the level of Chicago wheat stocks are the main factors explaining the current basis. A negative coefficient on national commercial stocks may be an indication that the variable is a proxy for hard wheat stocks. Changes in these stocks result in opposite movements in the demand for soft wheat (Chicago) stocks and hence the Chicago (soft wheat) basis.While the estimated model “explains” a relatively high proportion of overall basis variation, it is not entirely satisfactory in predicting the direction of change in the basis. It also exhibits some downward bias when used to make forward predictions of the magnitude of the basis.Mary of the modelling problems encountered during the analysis probably stem from incorporating intra- and inter-seasonal basis movements into the one model. A more productive approach to an analysis of this type would be to concentrate on estimating and predicting only one type of basis movement rather than both.

Inventories and Short-Run Output Stabilization

Empirical studies of the role played by production-smoothing in determining inventory behavior have accounted for the production-smoothing effect by the introduction of seasonal dummy variables, and/or the change in output as additional variables in the inventory equation. Lovell [7, 192-93] has reviewed the empirical results of the studies by Johnston, Modigliani and Sauerlender, and Mills, and also reported the results of his own investigation of the production-smoothing effect. The results of these studies lend very weak support to the empirical usefulness of the hypothesis. On the other hand, Ghali [4, 149-57] has produced some evidence strongly supporting the role of production smoothing. In this paper a simple model of inventory behavior is developed under the assumption that the cost functions facing the firm are quadratic. The model includes both the production-smoothing and the buffer-stock motives for holding stocks of finished goods. The model is applied to monthly data for six industries and to ten pooled time seriescross section samples on one of the industries. The results lend strong support to the effect of production-smoothing on inventory behavior. In the concluding section it is shown that the flexible accelerator model of inventory behavior is a special case of the production-smoothing model developed. Modigliani [10] * A summary of the results reported in this paper was presented at the First International Symposium on Inventories, Hungarian Academy of Science, Budapest, August, 1980. I am indebted to a referee of this Journal, and to my colleagues Edwin Fujii and Burnham Campbell for valuable comments and suggestions.

The Durbin-Watson Test for Serial Correlation: Bounds for Regressions with Trend and/or Seasonal Dummy Variables

The Durbin-Watson Test for Serial Correlation: Bounds for Regressions with Trend and/or Seasonal Dummy Variables

A Test of the Capacity Pressure Hypothesis Within a Simultaneous Equations Model of Export Performance

Until recently economet-ric analysis of export behavior has proceeded on the basis of single equation models, and it generally has emphasized price and income demand elasticities.' The single equation approach also has been standard in investigating the effects of of domestic demand on export performance, a topic that has figured prominently at both the theoretical and empirical levels in studies of British exports.2 Most empirical studies of the relationship between the of domestic demand and exports (hereafter referred to as the pressure effect) have used time series regressions of export volume on some combination of export price, world market prices, domestic economic activity, world economic activity, and a measure of the on domestic capacity. These studies have supported the capacity hypothesis, finding that export sales are inversely related to the on domestic capacity. The diversion of export sales to the domestic market as domestic activity intensifies is often attributed to a lower profitability of exports, and from the accompanying discussion it may be viewed as an export supply phenomenon. These studies, however, employing single equation methods, do not permit structural estimation of the impact of capacity on supply. For this reason a simultaneous equations approach is especially appropriate for studying the role of capacity on exports. The model presented in this paper consists of simultaneous supply, demand, and market clearing relations.3 The demand function relates export volume (quantity) to export unit value (considered endogenous), a unit value of world trade, the value of world trade, and to seasonal dummy variables. The supply function relates export volume to export unit value, domestic price, domestic economic capacity, to measures of domestic capacity pressure, and to seasonal dummy variables. Quarterly data are used to test the model both for the United States and for the United Kingdom. Twostage least squares, utilizing estimation methods described by Fair (1970), are employed. The results indicate that changes in the level of capacity pressure, although not the level itself, have had a negative impact on American exports. No support for the capacity effect was detected in the British case. From the point of view of technique, the findings suggest that autocorrelated disturbances should be allowed for in the estimation procedure.

Estimating Models of Seasonal Market Response Using Dummy Variables

Researchers often suggest the usefulness of regression models of market response as aids to market decision making. Commonly, models which involve monthly, bimonthly, or quarterly data explicitly account for seasonal variations by the use of dummy variables. This note critically examines the use of dummy variables in seasonal models. The results of the examination indicate that although the commonly used seasonal shift model (seasonal dummy variables which influence only the intercept term of the model) may be successful in explaining the variance in market response, it may not explain it in a manner indicative of market behavior, and therefore may result in misleading implications for marketing decisions. Alternate seasonal specifications are discussed and statistical methods for selecting the appropriate model specification are considered.

Bemerkungen zum gebrauch von saison-variblen bei der schätzung von bistributed-lag-moiellen unter polynomialen restriktionen

This paper deals with the question how seasonal dummies influence the estimation of the lag distribution. It concerns mainly the Shiller estimator and only partly the Almon estimator. A result of Lovell for the case of estimation under linear restrivtions is used, since the Shiller estimator satisfies the corresponding conditions. The connection between the Shiller and the Almon estimator is shown. the influence of the “maximum lag” which the user fixes is examined for both approaches “with” and “without” seasonal dummies.

Seasonal Variation in Interest Rates

A survey of the empirical work involving interest rates indicates an almost universal use of seasonally unadjusted rates, apparently due to a conviction that seasonal factors are not important in interest rates. Studies by V. Kerry Smith and Richard Marcis (1972), Stanley Diller (1969) and William Gibson (1970), however, provide evidence showing that interest rates exhibit seasonal variation. Seasonality in interest rates merits careful consideration for at least two reasons. First, as noted by Gibson, An aim of the Federal Reserve System is to accommodate seasonal swings in the financial needs of trade, and the system tries to do this by removing seasonal fluctuations from interest rates. The seasonal variations remaining in interest rates suggest that the system is not wholly successful in these efforts. . (Gibson, 1970, p. 442). Second, from an econometric point of view, any model, such as a money demand model, which contains an interest rate as an independent variable should also contain seasonal dummy variables; otherwise, the coefficient of the interest rate variable may be both biased and inconsistent.' These issues underscore the importance of determining whether there is seasonality in interest rates. In this regard, it is interesting to note that the Federal Reserve does not report seasonally adjusted rates, apparently because the Board does not recognize the existence of a seasonal component. The purpose of this paper is to provide additional information regarding seasonality in interest rates. Unlike earlier studies, both daily and monthly data are analyzed. And, whereas Smith and Marcis employed spectral analysis to detect seasonality, ordinary least squares techniques with seasonal dummies are employed here. Using this technique, we find, in contrast to the findings reported by other investigators, no evidence of a significant seasonal component in monthly rates. Seasonality is, however, present in the daily rate. The plan of the remainder of the paper is as follows. In section II, the magnitude and the importance of seasonality in three interest rates are assessed. Section III contains a discussion of seasonal variation in a short-term rate employing daily as opposed to monthly data. A summary and the conclusions are presented in the last section.

An Examination of the Financing of Federal Home Loan Bank Advances: A Comment

In a recent note in this Review,' Jene K. Kwon and Richard Thornton argue that their empirical research . . supports the implication of our model; to wit, that Federal Home Loan Bank (FHLB) bonds are a substitute for Savings and Loan (S and L) (p. 98). It is the contention of this note that no such distinctive implication can be drawn. Rather, what Kwon and Thornton have shown is that credit market instruments are substitutes for S and L deposits and that FHLB bonds are viewed by the market as one category of credit market instruments. This is not a new finding and is not a sufficient basis on which their policy recommendations can stand. Kwon and Thornton use two measures of savings flows to FSLIC-insured S and L's, net new savings (NNS) and changes in gross savings (AS). They conclude that NNS performs better on the basis of the R2 S2 (SE ?), Durbin-Watson and t statistics (p. 98). We will therefore confine our comments to this form of their model. Kwon and Thornton use a linear version of a stock adjustment model to compare various interest rate S and L deposit rate differentials in a model containing disposable income, lagged NNS and seasonal dummy variables. In all, six equations are estimated (p. 98). Their results indicate that when changes in the rate of return on FHLB bonds minus the rate on S and L deposits is estimated (equation 1.1, p. 98) the regression coefficient is significant at the 1 per cent level using the t test. However, it is also the case that when the commercial paper rate minus the rate on S and L deposits is estimated in the next equation, the regression coefficient is also significant at the 1 per cent level (equation 1.2, p. 98). Furthermore, the three-month bill-S and L differential (equation 1.3, p. 98) and the nineto twelve-month bill-S and L differential (equation 1.4, p. 98) are statistically significant at the 5 per cent level. Aaa corporate S and L rate differential (equation 1.5, p. 98) and the time deposit S and L rate differential (equation 1.6, p. 98) are the only interest rate differentials which are not statistically significant. Examining those equations in which the interest rate differentials are statistically significant, the implications of the analysis seem clear. There is no distinctive implication that FHLB bonds are substitutes for S and L deposits, rather the implication is that open market instruments are substitutes for S and L deposits and that as the rate spread between instruments and S and L deposits increase, the net new savings to S and L's decrease. This is certainly not a new finding and while interesting, this hardly shows that FHLB bonds are materially different from any other open market instrument in their effect on S and L deposits. This is to be expected as it is to be doubted that FHLB bonds are somehow special when viewed by the market along with other market instruments. This note has demonstrated that the implications drawn by Kwon and Thornton from their work are too strong. important unanswered questions that need to be addressed are: (1) how important an effect does the competition of FHLB bonds have on S and L deposit flows?, and the related question, (2) given that FHLB bonds do compete in the market with deposits at S and L's, what is the actual net impact on S and L's of an increase in FHLB bond flotations for the purpose of making advances to S and L's? 2 That the Kwon-Thornton work does not allow us to draw any conclusion about either of these questions is clear from regressing NNS on NNS lagged one period (one of the independent variables in their regression equations) and the seasonal dummy variables. In their equation 1.1 which contains the FHLB bond S and L rate differential, disposable income, and the seasonal dummy variables in addition to the lagged NNS variable, the adjusted coefficient of determination (R2) is 0.630. When only the lagged value of the dependent variReceived for publication May 4, 1972. Revision accepted for publication October 18, 1972. * opinions expressed in this note do not necessarily reflect those of the Federal National Mortgage Association. I wish to thank Harry S. Schwartz, Raymond Lombra and Richard Marcis for making useful comments on an earlier draft of this note. 1Jene K. Kwon and Richard M. Thornton, The Federal Home Loan Bank and Savings and Loan Associations: Examination of the Financing of Federal Home Loan Bank Advances, this REVIEW, LIV (Feb. 1972), 97-100. 2Kwon and Thornton make a somewhat similar statement in An Evaluation of the Competitive Effect of FHLB Advances by Savings and Loan Associations, Journal of Finance, XXVI (June 1971), 699-712.

Testing for Fourth Order Autocorrelation in Quarterly Regression Equations

A test for fourth order autocorrelation in the error term of a regression equation estimated from quarterly data is described. The development draws on the finite sample results of Durbin and Watson and illustrates how their procedure for the first order case can be generalized. In the model y = X,B + u where X is a matrix of fixed regressors and u, = put-4 + Bt, an appropriate test statistic for Ho: p = 0 is the statistic d4 = {z- )2/z2 computed from the least squares regression residuals z = y - Xb. Bounds to the significance points of d4 are tabulated. Maximum likelihood estimation methods are described; these are equally appropriate when lagged values of the dependent variable appear among the regressors, and they provide asymptotic tests for general-autoregressive error structures, as well as for the special case ut = oe1u_-1 + 04ut4 - aLa4ut-5 + et. Examples from the empirical literature are presented. THE POSSIBILITY THAT the errors in a regression equation estimated from quarterly data possess fourth order autocorrelation was considered, among other things, in a recent paper [28], and a non-parametric test was proposed. Appropriate generalized least squares estimation methods were also described. In this paper we first present a more rigorous solution to the problem of testing for fourth order autocorrelation, which utilizes the approach introduced by Durbin and Watson [6 and 7]. We then describe non-linear estimation methods which simultaneously estimate the regression coefficients and the parameters of the simple fourth order or more general autoregressive error structures. The usual interpretation of the error or disturbance term in econometric models is that it represents the effect of omitted or unobservable variables on the dependent variable. The error term might thus be expected to display certain features of observed economic variables, in particular, when quarterly data are employed, seasonal variation. Equally, when seasonally unadjusted data are being employed in order that one may attempt to explain seasonal variation in the dependent variable, along with other types of variation, by means of explanatory economic or seasonal dummy variables, then the presence of non-systematic seasonal variation, or an incomplete accounting for seasonality by the regressors, will produce seasonal effects in the error term, with the possible consequence of fourth order autocorrelation. Thus we require a test for correlation not between the errors 1 Some of the results contained in this paper were reported in my paper Estimation and Tests for Quarterly Regression Equations with Autocorrelated Errors presented at the Second World Congress of the Econometric Society in Cambridge, September, 1970. I am grateful to David Hendry for comment and discussion and, in Section 3, for the use of his computer program, to Zvi Griliches and an anonymous referee for comments, to Andrew Tremayne for research assistance, and to M. I. Nadiri and Michael Parkin for supplying their data. Added in proof: After this was written an unpublished paper by H. D. Vinod entitled Generalization of the Durbin-Watson Statistic for Higher Order Autoregressive Processes was brought to my attention; this considers statistics similar to d4 for tests of higher order autocorrelation in the non

A STUDY OF THE BORROWING PATTERNS OF EIGHTEEN CLEVELAND-TERRITORY, FOURTH FEDERAL RESERVE DISTRICT MEMBER BANKS OVER THE PERIOD, 1961-1965*

Tiis DISSVRTATION INVOLVES a study of daily borrowing and reserve adjustments of 18 member banks in the Cleveland Federal Reserve District during the period, 1961 to 1965. Individual-bank data are studied as well as cross-sectional data for the 18 banks. A linear equation is estimated for the cross-sectional data using the following independent variables: deposit variability, reserve position, bank size (as a proxy for transaction costs), and various liquidity measures. A linear time-series equation is estimated also for each bank using deposits, interest rates, lagged borrowing, lagged reserve-position measures, and seasonal dummy variables as independent variables. A descriptive comparative-ratio analysis of each bank's reserve-adjustment behavior is also made. The study follows in the tradition of those of Goldfeld, De Leeuw, Goldfeld and Kane, Bryan and Carleton, and others who have attempted to estimate aggregate member-bank borrowing and reserve-adjustment behavior. The speed with which banks adjust reserves is studied with comparisons made between large and small banks. Except for a recent study by Bryan and Carleton, little has been published regarding daily reserve adjustments by individual banks. Knowledge about this behavior is important for a better understanding of the effectiveness of monetary policy. The cross-section results indicate that deposit variability is the most significant explanatory factor of borrowing for the 18 banks. The importance of this result lies in the fact that deposit variability is computed to measure unpredictable deposit flows around a given seasonal pattern of variability. This unpredictability of deposits is important for bank asset -managers. Moreover, in other studies of variability done primarily by individual Federal Reserve Banks, the regular calendar-related variability in deposits is not considered. In this study, the latter measure of variability proved highly insignificant in explaining borrowing. Other results of the study indicate that deposit variability itself is subject to influences such as deposit composition and bank size. Variability tends to decline with the percentage of time deposits, and large banks show less overall variability than small banks, because offsetting deposit flows characterize large banks. Bank size served as a proxy for transaction costs in the study and large banks were found to borrow less relative to deposits than small banks because borrowing from the Fed is typically more expensive for large banks than either selling securities or buying federal funds. Small banks appear to borrow more, since borrowing at the Fed is relatively less expensive than other sources of reserves. Results with individual-bank estimates indicate that large banks adjust reserves more rapidly than small banks and that interest rates are more significant for large banks in the adjustment process. Lagged reserves are significantly and inversely related to borrowing for each bank studied. Seasonal borrowing and reserve-adjust-