Introduction

Okun’s Law suggests that the gap between an economy’s full employment output and its actual level of output increases by two percentage points for each percentage point the unemployment rate increases. It generally only applies to US employment and output. Okun’s “law” is really more of a rule of thumb than law. It’s one of the first concepts learned in any intermediate macroeconomics course, and one of the more practical theories that have been proposed and supported by countless empirical studies. To better understand this concept, we’ll examine the logic behind Okun’s Law and actually do some coding in R to see if this “law” holds. After all, economic theories are almost useless if they can’t be supported by emperical evidence.

Okun’s Law

Calculating Okun’s Law is fairly simple. Mathematically, it can be expressed as

\[ \begin{equation} \label{eq:1} \frac{\bar{Y}-Y}{\bar{Y}} = 2(u - \bar{u}) \end{equation} \]

where \(\bar{Y}\) denotes the full level of employment and \(\bar{u}\) denotes the natural unemployment rate, and \((u-\bar{u})\) is what is considered the cyclical unemployment rate, the excess of the actual unemployment rate over the the natural rate of unemployment; essentially, the unemployment rate owing to the business cycle. The equation states that the difference between potential and actual output is equal to 2 times the cyclical unemployment rate.

As an example, let’s suppose that the natural rate of unemployment \(\bar{u}\) is \(4\%\), and that the level of full-employment output is $20 billion. If the actual unemployment rate is \(6\%\), two percentage points higher than the natural rate of employment, then cyclical unemployment \((u-\bar{u})\) is \(2\%\). Using this information, Okun’s law predicts that atual output \(Y\) will be \(4\%\) (\(2 \times 2\%\)) lower than full-employment level \(\bar{Y}\). And because the full-employment level of output is $20 billion, Okun’s law suggesets that actual output will be $20 billion \(\times\) \(4\%\) = $800 million.

For our purposes, it’s easier to express Okun’s law as

\[ \begin{equation} \label{eq:2} \frac{\Delta Y}{Y} = 3 - 2\Delta u \end{equation} \]

where \(\Delta Y / Y\) denotes the percentage growth rate of output and \(\Delta u\) is the change in the actual unemployment rate from one year to the next. This is useful in our analysis because this data is easily accessible. The level of full-employment output and the natural rate of unemployment of an economy are essentially theoretical concepts whose exact are not really known. The equation suggests that when unemployment is rising, actual output \(Y\) is growing more slowly than \(3\%\) per year, which is the average growth rate of full-employment output in the United States.

The Model

Our model will be a simple linear regression model in which the yearly percentage change in US real GDP will be our response variable and the yearly percentage change in US unemployment will be our explanatory variable.

\[ \begin{equation} \Delta Y \% = \beta_{0} + \beta_{1} \Delta u \end{equation} \]

If Okun’s Law holds, then the value of \(\beta_{0}\) and \(\beta_{1}\) should be 3 and 2, respectively.

The Data

Our real GDP data was taken from The Bureau of Economic Analysis website, where we used the data set Percent change from preceding period. For our unemployment data, we the Bureau of Labor Statistics provided a nice table that included the year and corresponding unemployment rate from 1947 to 2015. I took the liberty of cleaning the data so we only have the pertinent variables in our data set, which you can download here if you wish to follow along.

year y.pct.chg unemp.chg
1948 4.1 -0.1
1949 -0.5 2.1
1950 8.7 -0.6
1951 8.1 -2.0
1952 4.1 -0.3
1953 4.7 -0.1
1954 -0.6 2.6
1955 7.1 -1.1
1956 2.1 -0.3
1957 2.1 0.2
1958 -0.7 2.5
1959 6.9 -1.3
1960 2.6 0.0
1961 2.6 1.2
1962 6.1 -1.2
1963 4.4 0.2
1964 5.8 -0.5
1965 6.5 -0.7
1966 6.6 -0.7
1967 2.7 0.0
1968 4.9 -0.2
1969 3.1 -0.1
1970 0.2 1.4
1971 3.3 1.0
1972 5.2 -0.3
1973 5.6 -0.7
1974 -0.5 0.7
1975 -0.2 2.9
1976 5.4 -0.8
1977 4.6 -0.6
1978 5.6 -1.0
1979 3.2 -0.3
1980 -0.2 1.3
1981 2.6 0.5
1982 -1.9 2.1
1983 4.6 -0.1
1984 7.3 -2.1
1985 4.2 -0.3
1986 3.5 -0.2
1987 3.5 -0.8
1988 4.2 -0.7
1989 3.7 -0.2
1990 1.9 0.3
1991 -0.1 1.2
1992 3.6 0.7
1993 2.7 -0.6
1994 4.0 -0.8
1995 2.7 -0.5
1996 3.8 -0.2
1997 4.5 -0.5
1998 4.5 -0.4
1999 4.7 -0.3
2000 4.1 -0.2
2001 1.0 0.7
2002 1.8 1.1
2003 2.8 0.2
2004 3.8 -0.5
2005 3.3 -0.4
2006 2.7 -0.5
2007 1.8 0.0
2008 -0.3 1.2
2009 -2.8 3.5
2010 2.5 0.3
2011 1.6 -0.7
2012 2.2 -0.8
2013 1.5 -0.7
2014 2.4 -1.2
2015 2.4 -0.9

It’s important to note the change in real GDP is expressed as in percentages, while the change in the unemployment rate is expressed as percentage points.

Let’s go ahead and plot our data.

library(ggplot2)
library(ggthemes)
okun <- read.csv('/Users/cyobero/Desktop/okun.csv')
okun <- na.omit(okun)

g <- ggplot(okun, aes(y=y.pct.chg, x=unemp.chg)) + geom_point(col='blue') + theme_economist()
g + xlab('∆ in Unemployment Rate (percentage points)') + ylab('%∆ in real GDP')

Our graph seems to support the idea behind Okun’s Law that as unemployment increases, potential output decreases. However, we have yet to confirm that a 1% increase in the cyclical unemployment rate results in a 2% decrease in output. We can do that by using a simple linear regression model.

okun.lm <- lm(y.pct.chg ~ unemp.chg, data = okun)
summary(okun.lm)
## 
## Call:
## lm(formula = y.pct.chg ~ unemp.chg, data = okun)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -2.9762 -0.8710  0.1512  0.9752  4.3993 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   3.2479     0.1722   18.86   <2e-16 ***
## unemp.chg    -1.7547     0.1592  -11.03   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.42 on 66 degrees of freedom
## Multiple R-squared:  0.6481, Adjusted R-squared:  0.6428 
## F-statistic: 121.6 on 1 and 66 DF,  p-value: < 2.2e-16

Recall that the formula for Okun’s Law we’ve been using was

\[ \begin{equation} \label{eq:3} \frac{\Delta Y}{Y} = 3 - 2\Delta u \end{equation} \]

According to our model estimates, our equation looks more something like

\[ \begin{equation} \label{eq:4} \frac{\Delta Y}{Y} = 3.2 - 1.8\Delta u \end{equation} \]

Our model isn’t too far off from Okun’s Law. Perhaps if we had included more observations, our parameter estimates might have been a lot closer to 3 and 2. Let’s go ahead and graph our fitted line.

okun$yhat <- okun.lm$fitted.values
okun.predict <- predict.lm(okun.lm, interval = 'confidence', level = .95, se.fit = TRUE)
okun$upr <- okun.predict$fit[, 3]
okun$lwr <- okun.predict$fit[, 2]
g <- ggplot(okun, aes(y=y.pct.chg, x=unemp.chg)) + geom_point(col='blue') + theme_economist()
g <- g + geom_line(aes(y=yhat), col='red') 
g <- g + xlab('∆ in Unemployment Rate (Percentage Points)') + ylab('∆% in real GDP')
g + geom_ribbon(aes(ymin=lwr, ymax=upr), alpha=.1)

Conclusion

Okun’s Law describes the relationship between the US’s change in unemployment and percent change in output. Okun suggests that the gap between an economy’s full employment output and its actual level of output increases by two percentage points for each percentage point the unemployment rate increases. It is an extremely useful concept in economics as it attempts to explain how unemployment can effect an economy’s potential output. Using real US unemployment and real GDP data taken from the BEA and BLS, we conducted an empirical analysis of Okun’s Law to determine whether it still holds. We concluded that it still does, despite our numbers being off by a few tenths. Whether you’re well-versed in economics or not, Okun’s Law is something that everyone should at least be familiar with.