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Contents : T he Logic of Multiple Regression based on Chapter 8 of Judd & McClelland (1989) DATA The data are the Industrial Production (IP) index and the number of Unemployed (UN) workers (in millions) for the 10 years from 1950 to 1959. For convenience the years have been relabeled from 1 to 10. Here are the data: YR 1 2 3 4 5 6 7 8 9 10 IP 113 123 127 138 130 146 151 152 141 159 UN 3.1 1.9 1.7 1.6 3.2 2.7 2.6 2.9 4.7 3.8 Logic of Multiple Regression 2 P redicting UN with IP We will first try to predict UN with IP because it seems reasonable that with higher Industrial Production there would be more jobs and hence less Unemployment. A: UN -0.035 + 0.021 IP C: UN 2.8 UN -0.035 + 0.021 IP 3.5 3 2.5 2 120 130 140 150 IP 160 Doesn't look too good! Let's look at the predictions and the errors. We will use the special notation UN.0.IP to represent the errors in order to indicate that these are the errors from a model predicting UN using a constant (X0) and one predictor (IP). 12/8/95 Logic of Multiple Regression YR 1 2 3 4 5 6 7 8 9 10 IP 113 123 127 138 130 146 151 152 141 159 3 UN 3.1 1.9 1.7 1.6 3.2 2.7 2.6 2.9 4.7 3.8 SSE(C) 8.38 PRE 0.098 UN 2.3 2.5 2.6 2.8 2.7 3. 3.1 3.1 2.9 3.3 UN.0 IP 0.8 -0.61 -0.89 -1.2 0.55 -0.29 -0.49 -0.21 1.8 0.55 SSE(A) 7.56 F*(1 8) 0.87 UN.0 IP 2 0.64 0.37 0.8 1.5 0.3 0.082 0.24 0.044 3.3 0.3 SSR 0.82 p 0.38 That is using IP reduces error in predicting UN (over a simple model) by only about 10% which is not surprising. So do NOT reject MODEL C! IP by itself is NOT a useful predictor of UN. 12/8/95 Logic of Multiple Regression 4 P redicting UN with YR Let's try predicting UN with YR. Perhaps there are consistent yearly changes in unemployment. But we wil do this step-by-step slowing taking UN apart into its component pieces. We will start with modeling UN with its mean. A: UN 2.8 C: UN 0 UN UN 2.8 4.5 4 3.5 3 2.5 2 YR 2 4 12/8/95 6 8 10 Logic of Multiple Regression YR 1 2 3 4 5 6 7 8 9 10 UN 3.1 1.9 1.7 1.6 3.2 2.7 2.6 2.9 4.7 3.8 5 UN 2.82 2.82 2.82 2.82 2.82 2.82 2.82 2.82 2.82 2.82 UN.0 0.28 -0.92 -1.12 -1.22 0.38 -0.12 -0.22 0.08 1.88 0.98 YR.0 -4.5 -3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5 4.5 UN.0 is interpreted as the amount by which unemployment was unexpectedly (relative to the mean) high or low in a given year. For example the value of UN.0 0.28 for YR 1 means unemployment was unexpectedly high by .28 million 280 000 workers in the first year of these data. YR.0 has a similar interpretation even though it may seem at first. The value of YR.0 -4.5 means that YR is "unexpectedly" low (relative to the mean for YR) in the first year. The question now becomes whether knowing whether YR.0 is unexpectedly high or low (i.e. early or late) can predict when UN.0 is unexpectedly high or low. 12/8/95

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