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Use the data in PHILLIPS for this exercise.(i) In Example 11.5, we estimated an expectations augmented Phillips curve of the form$$\Delta i n f_{t}=\beta_{0}+\beta_{1} u n e m_{t}+e_{t}$$where $\Delta i n f_{t}=i n f_{t}-i n f_{t-1} .$ In estimating this equation by OLS, we assumed that the supply shock, $e_{r}$ was uncorrelated with unem $_{r}$ If this is false, what can be said about the OLS estimator of $\beta_{1} ?$(ii) Suppose that $e_{t}$ is unpredictable given all past information: $$\mathrm{E}\left(e_{t} | i n f_{t-1},$ unem $_{t-1}, \ldots\right)=0$$ Explain why this makes unem $_{t-1}$ a good IV candidate for unem.(iii) Regress unem, on unem $_{t-1} .$ Are unem, and $u n e m_{t-1}$ significantly correlated?(iv) Estimate the expectations augmented Phillips curve by IV. Report the results in the usual formand compare them with the OLS estimates from Example 11.5.

(i) the OLS estimator of $\beta_{l}$ is considered biased and inconsistent (ii) if that means there is no correlation between $e_{t}$ and unem$_{t-1}$. This is one important condition which is fulfilled by a variable to be IV (lnstrumental Variable) . This makes unem $_{t-1}$ a good IV (iii)The coefficient of unem is $0.732$ (iv) The coefficient of unem is $-0.138$ statistically insignificant at 5$\%$

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Chapter 15

Instrumental Variables Estimation and Two Stage Least Squares

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part one. If unemployment is correlated with the error terms ls will be biased and inconsistent in estimating beta one. Mhm, yeah, yeah, Part two. If the conditional expectation of the error term on past inflation and unemployment rate is zero, then black unemployment rate is one correlated with the air on terms, which means the first leg of unemployment rate satisfied the first requirement for an instrumental variable. And yeah, yeah, US unemployment can work as an instrument in the equation where you have the change in inflation as a function of contemporary unemployment rate. Okay, The second requirement for unemployment mhm lack of unemployment rate to be a valid instrument for unemployment is that the lack of unemployment rates should be. It's officially correlated. Mhm, mhm. But you know, mhm. The regression on of unemployment on unemployment rate in the past period gives so based on the coefficient of past unemployment, you can conclude that there is a strong positive correlation between unemployment rate of this period and last period. Part four. The expectation of demented Phillips curve estimated by instrumental variable is the change in inflation of this period equals 0.69 minus 0.13 times unemployment rate of this period. Yeah. Oh, the I V estimate of beta one is much lower in magnitude than the old ls estimate, which is minus 10.543 And Beethoven is not statistically different from zero. But the old ls estimate is significant. We

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