Okun's Law - see, for example, Mankiw (1994, Chapter 2)-...

Okun's Law - see, for example, Mankiw $(1994,$ Chapter 2$)-$ implies the following relationship between the annual percentage change in real GDP, pcrgdp, and the change in the annual unemployment rate, cunem: $p c r g d p=3-2 \cdot$ cunem. If the unemployment rate is stable, real GDP grows at 3$\%$ annually. For each percentage point increase in the unemployment rate, real GDP grows by two percentage points less. (This should not be interpreted in any causal sense; it is more like a statistical description.) To see if the data on the U.S. economy support Okun's Law, we specify a model that allows deviations via an error term, pcrg$d p_{t}=\beta_{0}+\beta_{1}$ cunem $_{t}+u_{t}.$ (i) Use the data in OKUN to estimate the equation. Do you get exactly 3 for the intercept and $-2$ for the slope? Did you expect to? (ii) Find the $t$ statistic for testing $\mathrm{H}_{0} : \beta_{1}=-2 .$ Do you reject $\mathrm{H}_{0}$ against the two-sided alternative at any reasonable significance level? (iii) Find the $t$ statistic for testing $\mathrm{H}_{0} : \beta_{0}=3 .$ Do you reject $\mathrm{H}_{0}$ at the 5$\%$ level against the two-sided alternative? Is it a "strong" rejection? (iv) Find the $F$ statistic and $p$ -value for testing $\mathrm{H}_{0} : \beta_{0}=3, \beta_{1}=-2$ against the alternative that $\mathrm{H}_{0}$ is false. Does the test reject at the 10$\%$ level? Overall, would you say the data reject or tend to support Okun's Law?

Okun's Law - see, for example, Mankiw $(1994,$ Chapter 2$)-$ implies the following relationship between the annual percentage change in real GDP, pcrgdp, and the change in the annual unemployment rate, cunem:
$p c r g d p=3-2 \cdot$ cunem.
If the unemployment rate is stable, real GDP grows at 3$\%$ annually. For each percentage point increase in the unemployment rate, real GDP grows by two percentage points less. (This should not be interpreted in any causal sense; it is more like a statistical description.)
To see if the data on the U.S. economy support Okun's Law, we specify a model that allows deviations via an error term, pcrg$d p_{t}=\beta_{0}+\beta_{1}$ cunem $_{t}+u_{t}.$
(i) Use the data in OKUN to estimate the equation. Do you get exactly 3 for the intercept and $-2$ for the slope? Did you expect to?
(ii) Find the $t$ statistic for testing $\mathrm{H}_{0} : \beta_{1}=-2 .$ Do you reject $\mathrm{H}_{0}$ against the two-sided alternative at any reasonable significance level?
(iii) Find the $t$ statistic for testing $\mathrm{H}_{0} : \beta_{0}=3 .$ Do you reject $\mathrm{H}_{0}$ at the 5$\%$ level against the two-sided alternative? Is it a "strong" rejection?
(iv) Find the $F$ statistic and $p$ -value for testing $\mathrm{H}_{0} : \beta_{0}=3, \beta_{1}=-2$ against the alternative that $\mathrm{H}_{0}$ is false. Does the test reject at the 10$\%$ level? Overall, would you say the data reject or tend to support Okun's Law?

Key Concepts

Okun's Law

Okun's Law describes an empirical relationship between the change in the unemployment rate and the growth rate of real GDP. It illustrates how deviations in unemployment are statistically associated with deviations in GDP growth, serving as a rule of thumb in macroeconomics.

Linear Regression Model

A linear regression model is a statistical tool used to quantify the relationship between a dependent variable and one or more independent variables. It involves estimating an intercept and slope coefficients that best explain the observed variation in the data.

Estimation of Regression Coefficients

This concept involves using data to compute the intercept and slope of the linear regression model, which represent the expected value of the dependent variable when the independent variable(s) are zero and the magnitude of change in the dependent variable for a one-unit change in the independent variable, respectively.

Hypothesis Testing in Regression

Hypothesis testing in regression is used to determine whether the estimated coefficients differ significantly from certain hypothesized values. It forms the basis for making inferences about the population parameters underlying the model.

t-Statistic

The t-statistic is employed to test individual null hypotheses about regression coefficients. It measures the number of standard errors that the estimated coefficient is away from the hypothesized value, helping determine if the coefficient is statistically significant.

F-Statistic

The F-statistic is used for testing joint hypotheses about multiple coefficients simultaneously. It assesses whether the overall regression model significantly explains the variation in the dependent variable compared to a restricted model where some coefficients are set to their hypothesized values.

p-Value

The p-value provides the probability of obtaining a test statistic at least as extreme as the one observed if the null hypothesis is true. It is a key measure in hypothesis testing to determine whether there is sufficient evidence to reject the null hypothesis at a specified significance level.

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