Question 3: Decide To Use Non-Seasonally Adjusted Data On R

Question 3we Decide To Use Non Seasonally Adjusted Data On Retail Sal

Decide to use non-seasonally adjusted data on retail sales, adjusted for inflation. To begin with, we estimate an AR(1) model with observations on the annualized monthly growth in real retail sales from February 1972 to December 2000. We estimate the following equation: Table 5 shows the results from this model.

A) What do the autocorrelations show and why?

B) Suppose we add the seasonal lag of sales growth (the 12th lag) to the AR(1) model to estimate the equation: Table 6 presents the results. What do the autocorrelations show and why?

C) How can we interpret the coefficients in the model?

D) If retail sales grew at an annual rate of 5 percent last month and at an annual rate of 10 percent 12 months ago, the model predicts that retail sales will grow in the current month at an annual rate of?

Paper For Above instruction

Financial analysts and researchers often employ time series models to understand and forecast retail sales trends, considering the presence of seasonal patterns and autocorrelations. In this discussion, we analyze the initial AR(1) model and its enhancements in capturing the dynamics of retail sales growth, focusing on autocorrelation issues, model specification, and interpretability of estimated coefficients.

The initial model in question is an autoregressive model of order one, AR(1), where the dependent variable is the annualized monthly growth rate of real retail sales. This model aims to capture the dependence of current growth on its immediate past. The autocorrelations observed from the residuals of this initial model are critical in understanding whether the AR(1) specification sufficiently captures the underlying dynamics of the data. Typically, the autocorrelation function (ACF) of residuals from such a model should reveal no significant autocorrelation if the model is correctly specified. However, if persistent autocorrelations are detected—such as a significant autocorrelation at lag 12—it indicates the presence of seasonal effects not accounted for in the simple AR(1) model.

In the results presented in Table 5, the autocorrelations likely show significant spikes at seasonal lags, particularly lag 12, reflecting annual seasonal patterns in retail sales growth. These findings suggest that the model needs to incorporate seasonal lag variables to effectively model the series. This leads to the modification where a seasonal lag (the 12th lag) of sales growth is added to the model. As documented in Table 6, the inclusion of this seasonal lag often reduces autocorrelation at lag 12, indicating improved model specification. The autocorrelations after this addition should decay more rapidly, reflecting a more accurate capture of seasonal dependencies.

Interpreting the coefficients of these models involves understanding how past sales growth influences current growth. In the AR(1) model, the estimated coefficient on the lagged growth term indicates the persistence of growth — a positive coefficient suggests that higher growth in the previous period tends to be followed by higher growth in the current period. When seasonal lags are added, their coefficients quantify the seasonal influence; a significant positive coefficient at lag 12 implies that retail sales growth tends to be correlated with its value one year prior, emphasizing the importance of seasonal factors.

In the specific example where retail sales grew at 5% last month and 10% twelve months ago, the model's equation can be used to predict the current month's growth rate. Assuming the estimated coefficients from the model are known—say, the AR(1) coefficient and the seasonal lag coefficient—the forecast for the current month’s growth is computed by substituting these known past growth values into the model. For instance, if the coefficient on the previous month’s growth is 0.3 and the coefficient on the seasonal lag is 0.4, with last month’s growth at 5%, and twelve months prior at 10%, the predicted current growth rate would be computed accordingly, providing guidance for analysts on expected retail performance.

Regarding model validity and specification, the coefficients \(b_0\) and \(b_1\) should be statistically significant, with confidence intervals not including zero, indicating that they are reliably estimated. Tests such as t-tests for individual coefficients are used to assess their significance. If the model is mis-specified—for example, if autocorrelation remains after including seasonal terms or if residuals exhibit heteroskedasticity—then further steps are necessary. These include analyzing residuals with autocorrelation function (ACF) and partial autocorrelation function (PACF), testing for stationarity, and considering more complex models like ARMA or ARIMA that incorporate moving average components or differencing.

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