Applied Econometrics I
Exercise #1: The Taiwan Stock Market
The goal of this exercise is to show how to do basic regression analysis with very few variables. We will make use of the total.gdt GRETL data set which I have made available on my website.
THEORETICAL ANALYSIS:
First we need a little theory to guide us in our analysis.
Here is simple model showing how that a stock’s price is determined by certain fundamental economic variables and the force of equilibrium.
The most important relation is stock-bond arbitrage relation which states that if you decide to invest money in the stock market or the bond market, the return should be equal in equilibrium. This relation assumes many things. We ignore random factors. We ignore risk and liquidity premiums that investors may have. We make other assumptions that will be spelled out below.
Stock-Bond Arbitrage Relation:
where
nominal price of stock bought at time t
nominal stock price at time t+1 as expected at time t
cash dividend paid to stockholders at time t
nominal yield to maturity on bonds (having similar risk to stock) at time t
We can now do our analysis by doing a sophisticated rearrangement of the relation.
Write the equilibrium relation as
Now assume (for simplicity) that a long run “average” yield to maturity for bonds. In addition, we assume perfect foresight, which means . When there is no randomness explicitly in the model, perfect foresight corresponds to “rational expectations”. The above relation can be rewritten as
and after repeated substitution we have
Next, we assume that so that g is the expected growth rate of dividends. We will further assume that the growth of dividends is less than the nominal interest rate. That is, g < r.
This lets us write
which implies
where x = . This simplifies to
This shows that the price of stock depends on certain fundamental economic variables.
(1) Dt = the current nominal dividend paid on stock
(2) r = the average nominal yield to maturity on long term bonds
(3) g = the expected growth rate of dividends
Our theory shows us that a stock’s price will rise when current profits, Dt, are increasing; it will fall whenever nominal interest rates, r, rise, and it will increase when the expected trend in profitability, g, increases. Our empirical model of the stock market should at least try to include these three variables or some proxies to these variables.
EMPIRICAL ANALYSIS:
We are also limited by the fact that we do not have data on expectations. Our data set also does not include dividends or profitability. Most studies of the stock market would at least use such data. But, here we concentrate on the macroeconomic factors affecting the stock market.
Fortunately, we do have very complete data on nominal GDP, nominal interest rates, and the nominal stock price index. This should be enough for our purposes.
We assume a model like the following:
We assume the growth of nominal GDP affects the growth of the stock price index positively since profitability and growth in PY should be directly related. This means that β1 should be positive. We would expect that an increase in rt would tend to cause a reduction in the growth of the index St. Therefore, we expect that β3 is negative if our theory is correct. Finally, we add lagged values of the growth rate of St in order to capture the expectations effect. In other words, we use the past St’s to predict the current value of St. This makes the growth rate of St dependent on the past trends in PY and r. We could add more or less lags, not necessarily 2 lags. The important thing to remember is that the lags represent our expectations of future movements in PY and r based on past trends. This corresponds to our long run expectations of r and g in our theory.
Estimation Results:
Model 1: OLS estimates using the 95 observations 1979:3-2003:1
Dependent variable: gs
VARIABLE COEFFICIENT STDERROR T STAT 2Prob(t > |T|)
0) const 0.440891 0.195433 2.256 0.026495 **
39) gpy 1.41317 0.597364 2.366 0.020144 **
27) FIRM -0.354186 0.142962 -2.477 0.015098 **
40) gs_1 0.876714 0.104359 8.401 < 0.00001 ***
41) gs_2 -0.212865 0.100214 -2.124 0.036403 **
Mean of dependent variable = 0.0876266
Standard deviation of dep. var. = 0.401097
Sum of squared residuals = 4.88169
Standard error of residuals = 0.232897
Unadjusted R-squared = 0.677192
Adjusted R-squared = 0.662845
F-statistic (4, 90) = 47.2009 (p-value < 0.00001)
Durbin-Watson statistic = 2.00137
First-order autocorrelation coeff. = -0.0125385
Regression Diagnostics:
The results of this regression are very strong. There are only a few shortcomings to the results.
(1) which agrees with our theory, t-stat is significant
(2) which agrees with our theory, t-stat is significant
(3) which indicates that there is no unit root and expectations are long
lasting. t-stats are significant
(4) which is relatively high
(5) D-W statistic is close to 2.00, but it is biased towards 2 when there are lagged
dependent variables and is therefore unreliable.
Additional diagnostic testing is done below.
Test for Autocorrelation:
LM test for autocorrelation up to order 4 -
Null hypothesis: no autocorrelation
Test statistic: LMF = 3.01729
with p-value = P(F(4,82) > 3.01729) = 0.0224714
This test indicates that these is autocorrelation in the residuals.
Test for Heteroskedasticity:
White's test for heteroskedasticity -
Null hypothesis: heteroskedasticity not present
Test statistic: TR^2 = 8.47723
with p-value = P(Chi-Square(8) > 8.47723) = 0.388293
This test indicates that there is no simple heteroskedasticity present.
Tests for Stability of the Regression:
Chow test for structural break at observation 1990.1 -
Null hypothesis: no structural break
Test statistic: F(5, 85) = 1.60629
with p-value = P(F(5, 85) > 1.60629) = 0.167148
CUSUM test for parameter stability -
Null hypothesis: no change in parameters
Test statistic: Harvey-Collier t(89) = -0.964081
with p-value = P(t(89) > -0.964081) = 0.337617
Both of these tests indicate that the regression is stable and there is no apparent structural change.