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Today we wish to see how our model would have faired forecasting the past 20 values of GDP. Why? Well ask yourself this: How can you know where your going, if you don’t know where you’ve been? Once you understand please proceed on with the following post.

First recall the trend portion that we have already accounted for:

> t=(1:258)

> t2=t^2

> trendy= 892.656210 + -30.365580*t + 0.335586*t2

And that the de-trended series is just that- the series minus the trend.

dt=GDP-trendy

As the following example will demonstrate- If we decide to assess the model with a forecast of the de-trended series alone we may come across some discouraging results:

> test.data<-dt[-c(239:258)]

> true.data<-dt[-c(1:238)]

> forecast.data<-predict(arima(test.data,order=c(10,0,0),include.mean=FALSE),n.ahead=20)$pred

Now we want to plot the forecast data vs. the actual values of the forecasted de-trended series to get a sense of whether this is accurate or not.

> plot(true.data,forecast.data)

> plot(true.data,forecast.data,main=”True Data vs. Forecast data”)

Clearly it appears as though there is little to no accuracy with the the forecast of our de-trended model alone. In fact a linear regression of the forecast data on the true data makes this perfectly clear.

> reg.model<-lm(true.data~forecast.data)

> summary(reg.model)

Call:

lm(formula = true.data ~ forecast.data)

Residuals:

Min 1Q Median 3Q Max

-684.0 -449.0 -220.8 549.4 716.8

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -2244.344 2058.828 -1.090 0.290

forecast.data 2.955 2.568 1.151 0.265

Residual standard error: 540.6 on 18 degrees of freedom

Multiple R-squared: 0.06851, Adjusted R-squared: 0.01676

F-statistic: 1.324 on 1 and 18 DF, p-value: 0.265

> anova(reg.model)

Analysis of Variance Table

Response: true.data

Df Sum Sq Mean Sq F value Pr(>F)

forecast.data 1 386920 386920 1.3238 0.265

Residuals 18 5260913 292273

Now, is a good time to not be discouraged, but rather encouraged to add trend to our forecast. When we run a linear regression of trend on GDP we quickly realize that 99.7 of the variance in GDP can be accounted for by the trend.

> reg.model2<-lm(GDP~trendy)

> summary(reg.model2)

Call:

lm(formula = GDP ~ trendy)

Residuals:

Min 1Q Median 3Q Max

-625.43 -165.76 -36.73 163.04 796.33

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.001371 21.870246 0.0 1

trendy 1.000002 0.003445 290.3 <2e-16 ***

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 250.6 on 256 degrees of freedom

Multiple R-squared: 0.997, Adjusted R-squared: 0.997

F-statistic: 8.428e+04 on 1 and 256 DF, p-value: < 2.2e-16

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**The Dancing Economist**.

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