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Use the data in PHILLIPS for this exercise, but only through 1996 .(i) In Example $11.5,$ we assumed that the natural rate of unemployment is constant. An alternative form of the expectations augmented Phillips curve allows the natural rate of unemployment to depend on past levels of unemployment. In the simplest case, the natural rate at time $t$ equals$u n e m_{t-1}$ If we assume adaptive expectations, we obtain a Phillips curve where inflation and unemployment are in first differences: $$\Delta i n f=\beta_{0}+\beta_{1} \Delta u n e m+u$$Estimate this model, report the results in the usual form, and discuss the sign, size, and statistical significance of $\hat{\beta}_{1}$ .(ii) Which model fits the data better, $(11.19)$ or the model from part (i)? Explain.

(i) See video. Negative and significant main slope, t-test suggests one-for-one tradeoff (ii) Model part (i) better

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Chapter 11

Further Issues in Using OLS with Time Series Data

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part one. This is the estimated equation. In first differences, we need to comment on the size sign and significance level of the main coefficient. The estimate on the change in unemployment rate the size of beta one hat is about 0.842 and this is quite large. The estimate is negative, implying that there is a trade off between unemployment and inflation. Okay, regarding statistical significance, we can calculate the T statistic. You intake 0.842 divided by 0.314 you get a T value of minus 2.68 This is quite a large number. So this variable is significant in exact it is statistically different from zero. We can also test whether these variable is whether the coefficient is statistically different from minus one. And for this hypothesis test, we will do another calculation for the T start. I mean, you would take the estimate of beta one hat minus 0.842 minus minus one. That is plus one altogether divided by its standard error, and the T value is 0.5. So we are unable to reject the knob, which means beta one is not statistically different. from minus one and with a beta one of minus one. The trade off between inflation and unemployment is almost a one, for one are to. We will compare this model with estimation Equation 11.19 and the model from part one has higher our square and it just it are square. So this model is better than the 11.19 it is so because this model can explain more variation in the difference of inflation.

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