Business Forecast Analysis - Assignment Example

?Business Forecasting Report 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. Seasonality is very pronounced, presumably because the data reveals the effect of manufacturers building up inventories for the Christmas shopping season. Figure 1: The Absolute Values for Investment Prior to Decomposition These periodic fluctuations from first to fourth quarters are decomposed and isolated in Figure 2 below. 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 EXPONENTIAL -0.001281684 0.293408 -0.00053 -0.00437 0.996533 Fit for Invest01 from CURVEFIT, MOD_8 QUADRATIC 1.000413379 0.170529 0.717171 5.866527 4.28E-07 a. Dependent Variable: 50 Quarters Unhappily, this model does a poor job of matching actual investment values in the holdback period (Table 6 below), principally because the estimated intercept is unreliable. Table 6: Comparison of Actual and Predicted Values, Holdback Period Period Actual Investment Predicted from Model 2 51 3430 1,886 52 4130 1,777 53 3403 1,664 54 3414 1,547 55 3652 1,427 56 4511 1,303 57 3426 1,176 58 3437 1,045 3. Decomposition with Box-Jenkins ARIMA Revisiting the decomposition into seasonality, cyclical and trend components, the linear regression now explains nearly three-fourths of total variation in the manufacturing investment data (Table 8) and, based on the Table 7 summary of coefficients, yields the following fitted line: Ypredicted = 617.28 + 1.005 (Seasonality) – 0.169 (Trend) + 0.058(Cyclicality) Such a model, the adjusted R2 suggests, is superior to one employing dummy variables for seasonality. Table 7: Coefficients of Three-Variable Decomposed Model Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 617.282 757.121 .815 .419 Seasonality Indices 1.005 .176 .855 5.702 .000 Exponential trend -.169 .251 -.120 -.674 .504 Cyclical .058 .062 .129 .937 .354 a. Dependent Variable: 50 Months Test series Table 8: Proportion of Total Variance Explained Model Summaryb Model R R Square Adjusted R Square Std. Error of the Estimate 1 .850a .723 .703 418.835 a. Predictors: (Constant), Cyclical, Seasonality Indices, Exponential trend b. Dependent Variable: 50 Months Test series Table 9: ANOVA for Explained Variance, Decomposed Model ANOVAb Model Sum of Squares df Mean Square F Sig. 1 Regression 1.965E7 3 6550326.672 37.340 .000a Residual 7543170.451 43 175422.569 Total 2.719E7 46 a. Predictors: (Constant), Cyclical, Seasonality Indices, Exponential trend b. Dependent Variable: 50 Months Test series This time, the model hews more closely to the modeling data (Figure 7 and Table 10 below) in accounting for both the long-term trend and cyclicality. That the present model does not approximate the severe seasonal variations exhibited by the investment behaviour of the manufacturing sector cannot be helped since: a) by definition, the fourth quarter was historically so skewed; and b) short-term forecasts for, e.g. a year ahead are rather easily adjusted for known seasonality factors. Figure 7: Comparison of Predicted Y and Actual Historical Data for Investments By the same token and taking into account that manufacturing investment may still be at a cyclically low point, the quarterly forecasts in Table 10 below can be adjusted for the fact they overestimate the first three quarters of the year but underestimate the holiday season. Table 10: Comparison of Predicted Values for Holdback Periods   Investment Yp 3Q2004 3430 3568 4Q2004 4130 3615 1Q2004 3403 3682 2Q2004 3414 3751 3Q2004 3652 3860 4Q2004 4511 3880 1Q2004 N.A. N.A. 2Q2004 N.A. N.A. N.A. = Not available because the centered moving average causes the last two quarters to be lost. It is therefore possible to terminate further analysis at this point since one already has a parsimonious model that outputs Yp that approximates very closely the long-term trend, the observed industry cycle, permits adjustment for stable seasonal variation (and a more accurate forecast compared to model 2), and accounts for an eminently satisfying 70% of the variation in the 50-quarter data (compared to 20% for the first model). Hence, the sole value that a Seasonal ARIMA would add is analysis of random shocks (not evident in the long-term trend) and using differencing to de-trend the data. In fact, Figure 8 overleaf demonstrates that differencing by 1 does not reduce the mean to zero nor does an extended differencing by 2 (taking the first difference and deducting from the next in the series). Figure 8: Applying Differencing of 1 and 2 to the Investment Data Read More