For the U.S. economy, let gprice denote the monthly growth...

For the U.S. economy, let gprice denote the monthly growth in the overall price level and let gwage be the monthly growth in hourly wages. [These are both obtained as differences of logarithms: gprice $=\Delta \log (\text {price}) \text { and gwage }=\Delta \log (\text {wage}) .]$ Using the monthly data in WAGEPRC, we estimate the following distributed lag model: i. Sketch the estimated lag distribution. At what lag is the effect of gwage on gprice largest? Which lag has the smallest coefficient? ii. For which lags are the $t$ statistics less than two? iii. What is the estimated long-run propensity? Is it much different than one? Explain what the LRP tells us in this example. iv. What regression would you run to obtain the standard error of the LRP directly? v. How would you test the joint significance of six more lags of $g$ wage? What would be the $d f$ s in the $F$ distribution? (Be careful here; you lose six more observations.)

For the U.S. economy, let gprice denote the monthly growth in the overall price level and let gwage be the monthly growth in hourly wages. [These are both obtained as differences of logarithms:
gprice $=\Delta \log (\text {price}) \text { and gwage }=\Delta \log (\text {wage}) .]$ Using the monthly data in WAGEPRC, we estimate the following distributed lag model: i. Sketch the estimated lag distribution. At what lag is the effect of gwage on gprice largest? Which lag has the smallest coefficient? ii. For which lags are the $t$ statistics less than two? iii. What is the estimated long-run propensity? Is it much different than one? Explain what the LRP tells us in this example. iv. What regression would you run to obtain the standard error of the LRP directly? v. How would you test the joint significance of six more lags of $g$ wage? What would be the $d f$ s in the $F$ distribution? (Be careful here; you lose six more observations.)

Key Concepts

Joint Significance Testing and Degrees of Freedom

Joint significance testing is used to determine whether a group of lagged coefficients jointly contribute to the explanatory power of the model. This involves conducting an F-test that compares a restricted model (without the lags in question) to an unrestricted one. Adjustments to the degrees of freedom must account for the number of restrictions imposed and any reduction in sample size due to lag inclusion, influencing the critical values for the F-test.

Distributed Lag Model

A distributed lag model is an econometric tool that captures the effect of an independent variable on a dependent variable distributed over several time periods (lags). It estimates how current and past values of a predictor (such as wage growth) influence the current value of the outcome (such as price growth), allowing for dynamic adjustment processes in the relationship.

Log Difference Growth Rates

Using differences of logarithms to measure growth rates is common in economics as it approximates percentage changes while stabilizing variance. This method transforms variables such as the price level and wage into growth rates, making them more suitable for time series analysis by focusing on relative rather than absolute changes.

Lag Distribution and Coefficient Patterns

The estimated lag distribution refers to how the coefficients corresponding to various lags of an independent variable are arranged over time. This distribution helps in identifying which lag has the largest or smallest effect on the dependent variable, revealing the timing and strength of the relationship between the variables.

t-Statistics and Significance Testing

t-Statistics are used to test the statistical significance of individual coefficients in the regression model. When the t-statistic is less than a critical value (commonly two in absolute value for a 5% significance level), it suggests that the coefficient might not be statistically distinguishable from zero, indicating a lack of evidence that the corresponding lag effect is different from zero.

Long-Run Propensity (LRP)

The long-run propensity, obtained by summing all the lag coefficients, measures the cumulative effect of a permanent change in an independent variable on the dependent variable. It provides insight into the overall adjustment or stabilization effect over the long term and indicates whether the eventual impact diverges substantially from unity, which is critical in understanding persistent economic relationships.

Standard Error Estimation for LRP

To obtain the standard error of the long-run propensity directly, one must perform a regression that constrains or aggregates the individual lag coefficients, reflecting the cumulative intention. This approach accounts for the covariance among estimated coefficients and provides a measure of the precision of the overall long-run effect.

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