where \(inf_t\) is the annual inflation rate and \(unem_t\) is the unemployment rate.
This form of the Phillips curve assumes a constant natural rate of unemployment and constant inflationary expectations, and it can be used to study the contemporaneous tradeoff between inflation and unemployment.
The regression indicates a positive relationship (\(\hat{\beta}_1>0\)) between inflation and unemployment at 5% significance level.
There are some problems with this analysis that we cannot address in detail now. The Classical Linear Model assumptions do not hold. In addition, the static Phillips curve is probably not the best model for determining whether there is a short run tradeoff between inflation and unemployment. Macroeconomists generally prefer the expectations augmented Phillips curve, while we will see shortly.
6.2 Expectations Augmented Phillips Curve
Example 12.3 in Greene (2003), 5ed, Econometric Analysis.
A linear version of the expectations augmented Phillips curve can be written as
where \(\mu_0\) is the natural rate of unemployment, which we assume constant over time, and \(inf^e_t\) is the expected rate of inflation formed in year \(t-1.\)
The difference between actual unemployment and the natural rate is called cyclical unemployment, while the difference between actual and expected inflation is called unanticipated inflation.
If there is a tradeoff between unanticipated inflation and cyclical unemployment, then \(\beta_1<0.\)
The error term, \(e_t\), is called a supply shock by macroeconomists.
To complete this model, we need to make an assumption about inflationary expectations. Under adaptive expectations, the expected value of current inflation depends on recently observed inflation. A particularly simple formulation is that expected inflation this year is last year’s inflation:
\[
inf^e_t = inf_{t-1}
\]
Under this assumption, we can write the following empirical model:
\(\hat{\beta}_1<0\) indicates a tradeoff between cyclical unemployment and unanticipated inflation. But the effect is statistically insignificant.
Under expectations augmented Phillips Curve, one-point increase in \(unem\) lowers unanticipated inflation by about 0.1 of a point. We can contrast this with the static Phillips curve in Equation 6.1, where we found a positive relationship between inflation and unemployment.
res <-tibble(res_t = lm_phillips_aug$residuals,res_t1 =lag(lm_phillips_aug$residuals))lm_res <-lm(res_t ~ res_t1, data = res)summary(lm_res)
Call:
lm(formula = res_t ~ res_t1, data = res)
Residuals:
Min 1Q Median 3Q Max
-9.8694 -1.4800 0.0718 1.4990 8.3258
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.02155 0.17854 -0.121 0.904
res_t1 -0.42630 0.06355 -6.708 2e-10 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.531 on 199 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.1844, Adjusted R-squared: 0.1803
F-statistic: 44.99 on 1 and 199 DF, p-value: 2.002e-10
There is strong evidence of serial correlation in the residuals.
6.3 Remedies for AR errors
Example 12.5 in Wooldridge (2013), 5ed, Introductory Econometrics: A Modern Approach.
Example 19.2 in Greene (2003), 5ed, Econometric Analysis.
6.3.1 Prais-Winsten Estimation (Transformation)
Static model
Static model indicates positive autocorrelation.
res <-tibble(res_t = lm_phillips_stat$residuals,res_t1 =lag(lm_phillips_stat$residuals))lm_res <-lm(res_t ~ res_t1, data = res)summary(lm_res)
Call:
lm(formula = res_t ~ res_t1, data = res)
Residuals:
Min 1Q Median 3Q Max
-7.5887 -1.6354 -0.1089 1.2856 7.9793
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.04062 0.18009 -0.226 0.822
res_t1 0.64520 0.05349 12.062 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 2.553 on 199 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.4223, Adjusted R-squared: 0.4194
F-statistic: 145.5 on 1 and 199 DF, p-value: < 2.2e-16
library(prais) # install.packages("prais")data <- data %>%mutate(yrQ =as.yearqtr(paste(Year, qtr, sep="-")))pw_est_stat <-prais_winsten(lm_phillips_stat, data = data %>%tail(-2), index ="yrQ" )