Chapter5 UNDERSTANDING LINEAR DEPENDENCE: A LINK TO ECONOMIC MODELS_solutions University of Alabama EC 410

CHAPTER 5. UNDERSTANDING LINEAR DEPENDENCE: A LINK TO ECONOMIC MODELS SOLUTIONS by Wei Lin and Yingying Sun (University of California, Riverside) Exercise 1 We simulate 1000 observations of the process pt = 6.43 + 0.55pt−1 + εt and plot 100 observations. Compare Figures 1, 2, and 3 with Figures 5.2 and 5.3 in the textbook. The price oscillates around an average price of $14.3 in all these graphs. The time series of the simulated prices in Figures 1, 2 and 3 exhibit smooth dynamics similar to those of the time series in Figure 5.3, in contrast to the zig-zag behavior of the simulated price in Figure 5.2. This is due to the sign of the autoregressive parameter, which is positive, i.e. φ = 0.55. When the variance of the error term εt increases, the time series become noisier and more volatile, so that it tends to ‘hide’ the time dependence. However, the autocorrelation functions in the three Figures 4, 5 and 6 deliver the same message. The profile of the three ACF and PACFs is the same: a smooth decay of the autocorrelations towards zero in the ACFs, and only a significant spike, partial autocorrelation of order one, in the PACFs. Different variances in the error term do not affect the autocorrelation functions because the effect of the error variance in the numerator and denominator of the autocorrelation coefficients cancel each other out. Observe that these autocorrelation functions are similar to that in Figure 5.3 of the textbook. The main difference with the time series and the ACF and PCF in Figure 5.2 of the textbook is the sign of the autoregressive parameter. In Figure 5.2, the sign is negative (φ = −0.6), which produces the ziz-zag behavior of the time series and the alternating signs of the autocorrelation coefficients. 1 Gloria Gonz´alez-Rivera Forecasting For Economics and Business 2013 12.0 12.5 13.0 13.5 14.0 14.5 15.0 15.5 16.0 Simulated price with variance = 0.25 Figure 1: Simulated price pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 0.25 11 12 13 14 15 16 17 Simulated price with variance = 1 Figure 2: Simulated price pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 1 10 11 12 13 14 15 16 17 18 Simulated price with variance = 2 Figure 3: Simulated price pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 2 2 Gloria Gonz´alez-Rivera Forecasting For Economics and Business 2013 Figure 4: ACF and PACF of pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 0.25 Figure 5: ACF and PACF of pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 1 Figure 6: ACF and PACF of pt = 6.43 + 0.55pt−1 + εt with σ2 ε = 2 3 Gloria Gonz´alez-Rivera Forecasting For Economics and Business 2013 Figure 7: Time Series of Orange Prices in Florida (red) and in California (blue) Exercise 2 We download agricultural prices from the USDA website: The most recent report on monthly producer prices for oranges is dated 2011. The time series run from September 1991 to September 2011. In Figure 7 we plot the time series of prices for Florida and California. The vertical line on June 2003 separates the sample considered in the textbook from the updated sample. Overall, there seems to be a slight trend in both series; this trend was not obvious with the data ending on June 2003 so that we may suspect a non-stationary behavior in the overall updated series. This trend may be justified either because less land is allocated to this crop, hence reducing the supply, or/and there is more demand (population growth, preferences, substitution effects, etc.). Prices in Florida are lower than those in California, which may be due to supply/demand effects or/and different qualities and different markets. In Florida, the production cycle goes from November to June so there are not prices in the off-season of summer and early fall. On the contrary, California has a continuous cycle with prices peaking up during the off-season in Florida. In Figure 8, we present the autocorrelation functions of both series. Their profiles are very similar: the ACF decays slowly towards zero, and the PACF shows one very large and significant spike, a partial autocorrelation of order one with a value of approximately 0.80. In the PACF of the California time series, we also observe a second marginally significant spike. Overall, these functions are similar to that in Figure 5.3 of the textbook and, as a first step, we could entertain a model like pt = c+φpt−1+εt. The most distinctive feature is in the ACF and PACF of the Florida time series: there are significant spikes for displacements 11, 12, 13, 14. This is an indication of a seasonal cycle (dependence between observations that are 12 months apart, this is to say, December to December prices, January to January prices, etc.), wh