This empirical illustration is based on one of the founding articles on autocorrelation, namely Hildreth and Lu (1960). The data used in this study are time series data with 30 four-weekly observations from 18 March 1951 to 11 July 1953 on the following variables:
cons consumption of ice cream per head (in pints) income average family income per week (in US dollars) price price of ice cream (per pint)
temp average temperature (in Fahrenheit)
A graphical illustration of the data is given in Figure 4.3, where we see the time series patterns of consumption, price and temperature (divided by 100). The graph clearly sug- gests that the temperature is an important determinant for the consumption of ice cream, which supports our expectations.
k k
0.8
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0 10 20 30
Time Consumption
Temp/100 Price
Figure 4.3 Ice cream consumption, price and temperature/100.
The model used to explain consumption of ice cream is a linear regression model withincome,priceandtempas explanatory variables. The results of a first OLS regres- sion are given in Table 4.9. While the coefficient estimates have the expected signs, the Durbin–Watson statistic is computed as 1.0212. For a one-sided Durbin–Watson test forH0∶𝜌=0, against the alternative of positive autocorrelation, at the 5% level (𝛼=0.05)we havedL=1.21(T=30,K=4)anddU=1.65. The value of 1.02 clearly implies that the null hypothesis should be rejected against the alternative of positive autocorrelation. When we plot the observed values of cons and the predicted values according to the model, as in Figure 4.4, we see that positive (negative) values for the error term are more likely to be followed by positive (negative) values. Apparently, the inclusion oftempin the model is insufficient to capture the seasonal fluctuation in ice cream consumption.
The first-order autocorrelation coefficient in 𝜀t =𝜌𝜀t−1+𝑣t
Table 4.9 OLS results Dependent variable:cons
Variable Estimate Standard error t-ratio
constant 0.197 0.270 0.730
price −1.044 0.834 −1.252
income 0.00331 0.00117 2.824
temp 0.00345 0.00045 7.762
s=0.0368 R2=0.7190 R̄2=0.6866 F=22.175 dw=1.0212
k k
ILLUSTRATION: THE DEMAND FOR ICE CREAM 123
0.6
0.5
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0 10 20 30
Time
Consumption
Figure 4.4 Actual and fitted values (connected) of ice cream consumption.
is easily estimated by saving the residuals from the previous regression and running a least squares regression ofetonet−1 (without a constant).8This gives an estimate ̂𝜌=0.401 with anR2 of 0.149. The asymptotic test forH0∶𝜌=0 against first-order autocorrela- tion is based on√
T̂𝜌=2.19. This is larger than the 5% critical value from the standard normal distribution given by 1.96, so again we have to reject the null hypothesis of no serial correlation. The Breusch–Godfrey test produces a test statistic of(T−1)R2=4.32, which exceeds the 5% critical value of 3.84 of a Chi-squared distribution with one degree of freedom.
These rejections imply that OLS is no longer the best linear unbiased estimator for𝛽 and, most importantly, that the routinely computed standard errors are not correct. It is possible to make correct and more accurate statements about the price elasticity of ice cream if we choose a more efficient estimation method, like (estimated) GLS. The iter- ative Cochrane–Orcutt method yields the results presented in Table 4.10. Note that the EGLS results confirm our earlier results, which indicate that income and temperature
Table 4.10 EGLS (iterative Cochrane–Orcutt) results Dependent variable:cons
Variable Estimate Standard error t-ratio
constant 0.157 0.300 0.524
price −0.892 0.830 −1.076
income 0.00320 0.00159 2.005
temp 0.00356 0.00061 5.800
̂𝜌 0.401 0.2079 1.927
s=0.0326∗ R2=0.7961∗ R̄2=0.7621∗ F=23.419 dw=1.5486∗
8There is no need to include a constant because the average OLS residual is zero.
k k Table 4.11 OLS results extended specification
Dependent variable:cons
Variable Estimate Standard error t-ratio
constant 0.189 0.232 0.816
price −0.838 0.688 −1.218
income 0.00287 0.00105 2.722
temp 0.00533 0.00067 7.953
tempt−1 −0.00220 0.00073 −3.016 s=0.0299 R2=0.8285 R̄2=0.7999 F=28.979 dw=1.5822
are important determinants in the consumption function. It should be stressed that the statistics in Table 4.10 that are indicated by an asterisk correspond to the transformed model and are not directly comparable with their equivalents in Table 4.9, which reflect the untransformed model. This also holds for the Durbin–Watson statistic, which is no longer appropriate in Table 4.10.
As mentioned before, the finding of autocorrelation may be an indication that there is something wrong with the model, like the functional form or the dynamic specification.
A possible way to eliminate the problem of autocorrelation is to change the specification of the model. It seems natural to consider including one or more lagged variables in the model. In particular, we will include the lagged temperaturetempt−1. OLS in this extended model produces the results in Table 4.11.
Compared with Table 4.9, the Durbin–Watson test statistic has increased to 1.58, which is in the inconclusive region(𝛼=0.05)given by (1.14, 1.74). As the value is fairly close to the upper bound, we may choose not to reject the null of no autocorrelation. Apparently, lagged temperature has a significant negative effect on ice cream consumption, whereas the current temperature has a positive effect. This may indicate an increase in demand when the temperature rises, which is not fully consumed and reduces expenditures one period later.9