Problem 4 [12 marks Let yt be the monthly U.S. liquor sales. To adequately choos

Question

Problem 4 [12 marks Let yt be the monthly U.S. liquor sales. To adequately choose a forecasting model for yt, a researcher suggested and estimated two models. The first model, M1, is an autoregressive with three lags, a quadratic trend and 12 monthly dummies. The second model, M2, contains only the quadratic trend and the 12 monthly dummies, but the innovations are treated as an autoregressive of order three where Et is white noise WN(0,02) and Et is white noise WN(0,02). To estimate the first model, we use ordinary least square regression. The results are given below. OLS estimation Results: Model 1 Dependent variable: Y Current sample: 1969:1 to 1993:12 Mean of dep. Var Std. Dev. Of dep. Var Sum of squared residuals Variance of Residuals LM het. Test Durbin Watson 2.288 Jarque-Bera test 9.64919 Ramsey RESET =.034450 F (zero slopes)3563.34 Schwarz BIC -6.93009 Log likelihood 662.314 1.84537 .379193 .212350 ,750352E-03 .027393 .995061 Adjusted R-squared Estimated Standard Variable .512181 582289 252671 2.62478 698171 4 253121 857067e-03 313689E-03 2.73222 133841E-05 488936E-06 2.7373 4.16244 7.42559 The second model is estimated using Nonlinear least squares and the results are in the following

Explanation / Answer

- The trend is defined as the long term movement in a time series, reflecting the underlying level of the series. Cycle on the other hand is a pattern that exists when data exhibit rises and falls that are not necessarily having a fixed period.  

- The difference in two models in terms of modelling the trend and cycle in the series is in model 1 innovations are modelled in order of moving average of order n but in model 2 innovations are modelled as a n auto regressive component of order 3..

- Both the models have similar predictive power so I would use model 2 as it incorporates non linear least square estimation

- With a deterministic linear trend the uncertainty associated with long run forecast is limited to the variation in statitonery deviations from that trend but in contrast with a stochastic trend the greater the forecasting horizon greater is the uncertainty in associated with the forecast. So deterministic linear trend wont greatly affect long run or short run forecasts.

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