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Choosing the Appropriate Time-Series Model

The following guidelines are used to determine the most appropriate model depending on the need:

1. Understand the investment problem. This is followed by choosing the initial model.
2. Plot the time series to check for covariance stationarity. Observe if there is seasonality, a significant shift, a linear trend, or an exponential trend. Any trend in the model signals that the time series is covariance stationary.
3. In case of a linear or an exponential trend, take the following steps:
• Plot the time series to determine if the trend is linear or exponential: If the plot is a straight line with an upward or downward slope, use a linear trend model. On the other hand, if the plot is a curve, use a log-linear trend model.
• Estimate the trend.
• Calculate the residuals.
• Test for serial correlation in the residuals using the Durbin-Watson test.
• If there is no significant serial correlation of the residuals, then the model is suitable for forecasting.

4. On the other hand, serial correlation in the residuals calls for a more complicated model such as an autoregressive model. If the data has serial correlation, recheck it for covariance stationary before running the AR model. Transform the nonstationary data in the following ways:

• First difference the time series if the data has a linear trend.
• Take the natural log of the data with an exponential trend before first differencing.
• Estimate different time-series models before and after the shift in the instance the time series shifts significantly during the sample period.
• Include seasonal lag for data with significant seasonality.

5. Model the transformed time series with an autoregressive model, AR (1) model.

• If there are serial correlations in the residual of the AR (1) model, then try an AR (2) model. Seasonal lags are added until there are no serial correlations.

6. Test if residuals have autoregressive conditional heteroskedasticity (ARCH).

• Regress the squared residuals from each period on the prior period squared residuals.
• Test whether the resulting coefficient is significantly different from 0.
• If the slope coefficient is statistically different from 0, the series shows an ARCH (1) effect and the need to correct it for heteroskedasticity. This can be done by the use of generalized least squares.
• The model is suitable for use if the slope coefficient is not significantly different from zero.

7. Test the forecasting performance of the out-of-sample forecasting performance and compare this to the in-sample performance of the RMSE.

Question

Consider an AR(1) model used to forecast quarterly retail sales of a certain company based on 200 observations.

$$\text{x}_{\text{t}}=\text{b}_{0}+\text{b}_{1}+\epsilon_{\text{t}}$$

The residual autocorrelations relating to a certain year are as presented in the following table:

$$\small{\begin{array}{c|c} \textbf{Lag} & \textbf{Autocorrelation} \\ \hline1 & -0.0832 \\ \hline2 & -0.0701 \\ \hline3 & 0.0891 \\ \hline4 & 0.9379\\ \end{array}}$$

The most appropriate statement regarding the AR(1) model at the 5% significance level is that:

1. the AR(1) model is sufficient.
2. the AR(1) model is misspecified and not appropriate for use.
3. the AR (1) model’s error terms are not serially correlated, and thus the model is correctly specified.

Solution

To check whether the model is correctly specified at the 5% level of significance,

$$\text{Standard error}=\frac{1}{\sqrt{\text{T}}}=\frac{1}{\sqrt{200}}=0.0707$$

We can compute the t-statistic for lag one as:

$$\text{t}_{\text{Statistic}}=\frac{\text{Residual Autocorrelation}}{\frac{1}{\sqrt{\text{T}}}}$$

$$\text{t}_{\text{Statistic}}=\frac{-0.0832}{0.0707}=-1.1768$$

Calculating the t-statistic for each lag, we obtain:

$$\small{\begin{array}{c|c|c|c} \textbf{Lag} & \textbf{Autocorrelation} & \textbf{Autocorrelation} & \textbf{T-statistic}\\ \hline1 & -0.0832394 & -0.0832 & -1.1768 \\ \hline2 & -0.0701264 & -0.0701 & -0.9915\\ \hline3 & 0.08908331 & 0.0891 & 1.26025 \\ \hline4 & 0.93786911 & 0.9379 & 13.2659\\ \end{array}}$$

There are 200 observations and two parameters, $$\text{b}_{0}$$ and $$\text{b}_{1}$$ to be estimated. Thus 198 (200-2) degrees of freedom. The critical t-value at the 5% significance level with 198 degrees of freedom is 1.97.

The t-statistics of the first three lagged autocorrelations are less than the critical value at the 5% significance level. We can therefore conclude that none of the first three lagged autocorrelations is significantly different from zero.

However, notice that the t-statistic for the 4th lag autocorrelation is greater than the critical value. We thus reject the null hypothesis that the 4th lag autocorrelation is zero. The conclusion is that there is seasonality in the time series. This implies that the AR(1) model’s error terms are serially correlated, and thus the model is misspecified and not appropriate for use.

LOS 3 (o) Determine an appropriate time-series model to analyze a given investment problem and justify that choice.

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