Thursday, January 6, 2011

How Much Increase does the Dow need to gain employment?

The Dow being up is big news to many people in America. It signifies a strengthening economy, and a possible end of hard times for millions of people.

However, the Dow Jones goes up annually. On January 2nd, 1929, the DJI closed at 307.01. January 3rd, 2011, the DJI closed at 11,670.75. Despite a stretch in time that consisted of the Great Depression and the recent recession, the DJI has managed to climb at an annual rate of 4.54%, out-pacing inflation at 3.16%.

So, it is clear, over time, the Dow is a good, though not risk-free investment for an individual. A person who put the equivalent of \$1 in 2011 dollars into the DJI in 1929, the person would have \$2.97 now.

How does this affect main street, however? Namely, how is the annual performance of the Dow reflective on unemployment rates?

While one immediately assumes a growth in the Dow equals growth in employment, that is not necessarily the case. Though it is a positive sign for growth, there needs to be a level fit to meet an ever growing workforce, but how does one determine this?

Now, understanding the behavior of the Dow is easy, it generally follows a curve at the rate a*(1+r)^n, where a = starting value, r = rate of growth, and n = years since the starting point.

So first, to get a feel for the data, I elected to view the growth rate of the dow as a function of two variables: years since 1970, and unemployment rate of the year, found in this table, with fields of year, year since 1970, average dow jones index for the year, and average unemployment rate.

My first equation is as follows:

> summary(nlsdow)

Formula: dow ~ c * exp(a * yr70 + b * unemp) + c

Parameters:
Estimate Std. Error t value Pr(>|t|)
a 0.091785 0.004272 21.483
b -16.314535 2.978873 -5.477 3.21e-06 ***
c 951.696519 134.844962 7.058 2.38e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1034 on 37 degrees of freedom

Number of iterations to convergence: 8
Achieved convergence tolerance: 8.044e-06

Or, dow = 951.7*exp(0.091785*(yr-1970)-16.315*u_rate). Logical, the more years out, the more the DJI grows, and high unemployment is a good sign of a bad economy.

Now, we have an idea of the form for regression using the unemployment rate as a dependent variable (after all, unemployment as a lagging indicator means it is a better candidate for a dependent variable). Since we see the initial equation takes the form Y = a*exp(b*X1+c*X2), we can turn this into b*X1+c*X2 = ln(Y/a) = ln(Y) - ln(a) = ln(Y) + A, where A = -ln(a).

Solving for X2 gives us [ln(Y) + A - b*X1]/c, and of course we can change this into u_rate = p*ln(DJI) + q*(year since 1970) + r, where p = 1/c, q = -b/c, and r = -ln(a)/c.

So, simply put, one can regress for the unemployment rate by regressing against the natural log of the Dow Jones Index and the year since the particular starting point (in this case, 1970).

Fortunately, transforming a variable array is easy in R. To write a variable in log space, simply enter:

> lnDow<-log(dow)

You are now all set to run the regression. Let's try it:

> lmU<-lm(unemp ~ yr70 + lnDow)
> summary(lmU)

Call:
lm(formula = unemp ~ yr70 + lnDow)

Residuals:
Min 1Q Median 3Q Max
-0.0184135 -0.0058837 0.0002107 0.0038651 0.0209755

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.3038799 0.0347165 8.753 1.52e-10 ***
yr70 0.0028846 0.0004947 5.831 1.06e-06 ***
lnDow -0.0376083 0.0055375 -6.792 5.38e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009397 on 37 degrees of freedom
Multiple R-squared: 0.5931, Adjusted R-squared: 0.5711
F-statistic: 26.96 on 2 and 37 DF, p-value: 5.976e-08

I see a lot of highly significant coefficients. Though the R-squared leaves a bit to be desired, there seems to be a good deal being said through this function.

The coefficients we care about, though, are for yr70 and lnDow. To get a point estimate of the growth rate needed to sustain employment, we need to know how much lnDow has to increase to equal the coefficient of yr70, and effectively balance it out to keep unemployment stable. Simple algebra says that this means:

.0028846(yr70) = .0376083(lnDow),

so a delta of one year means we solve

.0376083(delta_lnDow) = .0028846, or delta_lnDow=.0767.

To find the point-estimate percentage increase needed, we simply solve exp(.0767) - 1 = 7.972%.

While high, the good news is, the DJI achieved this mark 17 of 39 years in the sample, and averages 6.5% growth a year from the average of 1970 to the average of 2009, including the downturn after 2007.

So if you were to ask me what my break-even point is for a "main street approved" economy is? Watch for that DJI to crack 8% growth during the year.