# Quants Question

The table below shows the autocorrelations of the lagged residuals for the first differences of the natural logarithm of quarterly motorcycle sales that were fit to the AR(1) model: (ln salest − ln salest − 1) = b0 + b1(ln salest − 1 − ln salest − 2) + åt. The critical t-statistic at 5% significance is 2.0, which means that there is significant autocorrelation for the lag-4 residual, indicating the presence of seasonality. Assuming the time series is covariance stationary, which of the following models is most likely to correct for this apparent seasonality? Lagged Autocorrelations of First Differences in the Log of Motorcycle Sales Lag Autocorrelation Standard Error t-Statistic 1 −0.0738 0.1667 −0.44271 2 −0.1047 0.1667 −0.62807 3 −0.0252 0.1667 −0.15117 4 0.5528 0.1667 3.31614 A) (ln salest − ln salest − 4) = b0 + b1(ln salest − 1 − ln salest − 2) + åt. B) (ln salest − 1 − ln salest − 4) = b0 + b1(ln salest − 1) − b2(ln salest − 4) + åt. C) ln salest = b0 + b1(ln salest − 1) − b2(ln salest − 4) + åt. D) (ln salest − ln salest − 1) = b0 + b1(ln salest − 1 − ln salest − 2) + b2(ln salest − 4 − ln salest − 5) + åt.

Maybe this’ll work better: The table below shows the autocorrelations of the lagged residuals for the first differences of the natural logarithm of quarterly motorcycle sales that were fit to the AR(1) model: (ln sales_t - ln sales_t-1) = b0 + b1(ln sales_t-1 - ln sales_t-2) + e_t. The critical t-statistic at 5% significance is 2.0, which means that there is significant autocorrelation for the lag-4 residual, indicating the presence of seasonality. Assuming the time series is covariance stationary, which of the following models is most likely to correct for this apparent seasonality? Lagged Autocorrelations of First Differences in the Log of Motorcycle Sales Lag Autocorrelation Standard Error t-Statistic 1 -0.0738 0.1667 -0.44271 2 -0.1047 0.1667 -0.62807 3 -0.0252 0.1667 -0.15117 4 0.5528 0.1667 3.31614 A) (ln sales_t - ln sales_t-4) = b0 + b1(ln sales_t-1 - ln sales_t-2) + e_t. B) (ln sales_t-1 - ln sales_t-4) = b0 + b1(ln sales_t-1) - b2(ln sales_t-4) + e_t. C) ln sales_t = b0 + b1(ln sales_t-1) - b2(ln sales_t-4) + e_t. D) (ln sales_t - ln sales_t-1) = b0 + b1(ln sales_t-1 - ln sales_t-2) + b2(ln sales_t-4 - ln sales_t-5) + e_t.

As it is covariance stationary, we shouldn’t need to muck about with first differences. So C) should be fine to correct for the seasonality.

The first differences of the natural log is covariance stationery here. I would say D).

D

D. (ln sales_t - ln sales_t-1) = b0 + b1(ln sales_t-1 - ln sales_t-2) + b2(ln sales_t-4 - ln sales_t-5) + e_t Please tell me this is correct. I am not going back to Quant again.

D is the correct answer can someone explain please why? I cannot understand the answer. Your answer: C was incorrect. The correct answer was D) (ln salest − ln salest − 1) = b0 + b1(ln salest − 1 − ln salest − 2) + b2(ln salest − 4 − ln salest − 5) + åt. Seasonality is taken into account in an autoregressive model by adding a seasonal lag variable that corresponds to the seasonality. In the case of a first-differenced quarterly time series, the seasonal lag variable is the first difference for the fourth time period. Recognizing that the model is fit to the first differences of the natural logarithm of the time series, the seasonal adjustment variable is (ln salest − 4 − ln salest − 5). why are we taking ln sales t-4 - ln sales t-5?

Here you model (ln sales_t - ln sales_t-1). So think (ln sales_t - ln sales_t-1) = V_t. What you see is that V_t is correlated with V_t-4. So you model should be V_t = b0 + b1 x V_t-1 + b2 x V_t-4 + e_t. If you replace the various V_t by their expression, you find answer D). Hope this helps.