Business Forecasting Report 1. Scrutiny for Seasonal Effects, Trends and Cycles Visual inspection of the unadjusted manufacturing sector investment in the UK from the first quarter of 1994 and on for the next 49 quarters makes it plain that: There is a long-term cycle that peaked at the end of 1998 and reached bottom in the first quarter of 2004…
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Figure 2: Seasonality Indices, Investment In turn, the polynomial line for cyclicality is shown below. Figure 3: Cyclical Nature of the Investment Data Series 2. Modifying Analysis with Dummy Variables Since seasonality is very pronounced, we first try fitting a regression line with the categorical variable for quarter coded as “1” for the first quarter, “2” for the second quarter, and so on. Such a model, we see from Table 1 below, would explain only about 20.4% of the total variation for UK manufacturing investment during the 50 months under review. Next, Table 2 shows that the ratio of mean variation explained by the regression to unexplained variation (or residual) error) is high enough that we can reject at ? < 0.005 the null hypothesis that such a ratio could be zero. Table 1: Model Summary for Dummy Variable Method and Linear Trend Cycle Model Summary R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 df2 Sig. F Change 0.472 0.223 0.204 685.9345 0.222684 12.03206 1 42 0.001221 a. Predictors: (Constant), Dummy variable for period Table 2 ANOVAb for Overall Fit Model Sum of Squares df Mean Square F Sig. 1 Regression 5661157.641 1 5661157.641 12.032 .001a Residual 1.976E7 42 470506.123 Total 2.542E7 43 a. Predictors: (Constant), Dummy variable for period b. Dependent Variable: 50 Quarters We see then from Table 3 (overleaf) that the fitted line can be calculated (predicted) as: Ypredicted = 3616 + 320.83 (Quarter) Table 3: The Coefficients for the Bivariate Regression with Dummy Variable for Seasonality Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 3616.000 253.298 14.276 .000 Dummy variable for period 320.827 92.491 .472 3.469 .001 a. Dependent Variable: 50 Quarters Without accounting for the linear trend, a single dummy variable for season can only move around the overall mean. It inflates expectations beginning the first quarter of 2004 and all the way through to the holdback period of the last quarters in the data series. Figure 4: Predicted Y When Only Dummy Variable for Seasonality is Accounted For An exponential trend line (Fig. 5 overleaf) better matches the fact that the investment series is at the trough of a long-term cycle. This reduces the errors when predicting the near term, the eight quarters of the holdback period. Figure 5: The Exponential Trend Line Figure 6 (overleaf) now includes the estimate of cyclical impact with a quadratic, one-peak line. Given the coefficients reported in Table 5 overleaf, the fitted model is: Ypredicted = 3.66 - 0.0005 (Trend) + 0.72 (Cyclical) Table 4 below reports that this model explains nearly half (49%) of the total variance in manufacturing investment during the period. Table 4: Model Summary for Trend Line and Cyclical Component R R Square Adjusted R Square Std. Error of the Estimate Change Statistics R Square Change F Change df1 df2 Sig. F Change 0.7169 0.5139 0.4932 570.1612 0.5139 24.8449 2 47 0.00000004 Figure 6: Accounting for the Cyclical Component: A Quadratic Line with One Peak Assumed Table 5: Coefficients for the Model Comprising Trend and Cyclical Components Coefficientsa Unstandar-dized Coefficients Standar-dized Coefficients t Sig. Std. Error Beta (Constant) 3.655649926 1038.629 0.00352 0.997207 Fit for Invest01 from CURVEFIT, MOD_8
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