I looked at the effect of interest rates (three month t ...



Effects of Interest Rates, GDP and Corporate Profits to Stock Market Returns – Further Evidence of the “Random Walk” of Stock Prices

In an effort to learn more about the impact of certain economic variables on stock market returns, I chose to analyze the effect interest rates, GDP and corporate profits[1] have on the stock market. Early on in my analysis, I confirmed the well-known fact that predicting stock market returns is a difficult task. I looked separately at long term t-bill rates together with the other independent variables on the stock market.

I used the following variables in my analysis:

• the S&P 500 Return

• the 3-month treasury bill

• the 20/30 year treasury bill

• GDP

• Percentage change in GDP

• Corporate profits/ GDP

I have tracked these variables quarterly from 1978 – 1998.

My hypothesis is that stock market returns would tend to:

• Increase as GDP increases

• Increase as interest rates decrease (when interest rates are lower one would expect people to increase investments in the stock market). Investors increase their investment in the stock market in the expectation of higher future profits.

• Increase as Corporate Profits/GDP increase

My hypotheses were:

H0: ß t-bill rate = ß GDP = ß corporate profits/GDP = 0

Ha: ß t-bill rate = ß GDP = ß corporate profits/GDP ≠ 0

In the national income accounts, GDP is equal to wages, profits, interest and rent. If each of these four components, as a share of GDP, remain constant over time, the percent change in GDP will be equal to the percent change in corporate profits. Therefore stock price is a function of GDP (I would expect to see a significant positive result when GDP is regressed against the S&P500 return). If corporate profits increase as a share of GDP, the stock market may increase faster than GDP. Interest rates are often used as a predictor of future profitability. This is why I have used the three variables, interest rates, GDP and corporate profits as a percent of GDP, in my model. I would argue that stock prices are a function of corporate profits; as corporate profits increase in value, the value of stocks will also increase. People are more willing to buy stocks if companies are more profitable.

Stock prices should grow with GDP but in regression, stock prices will also grow with any variable correlated with GDP that also grows over time. As a solution to this problem, I have included a “time dummy” variable to pick up the correlation of stock returns with time and the input of any variable correlated with time.

In spite of these hypotheses regarding expected effects of the various predictor variables, it is well known that predicting changes in the stock market is quite difficult. With this in mind, I would not expect to see very significant results from my regression models when looking at S&P Return as the response variable.

Basic Regression

My basic regression model shows the effect three predictor variables: short-term interest rates, GDP and corporate profits as a percent of GDP have on the response variable: S&P 500 Return (i.e., the percentage change in the S&P 500 – in this case, on a quarterly change basis). I also look at this regression using a long-term interest rate to see the different impact this would have (the effect was minimal and I therefore do not show these regressions separately). The most effective regression model uses S&P Returns rather than S&P Index results; using return data rather than index data accounts for the fact that it is more accurate to see the relative influence of the change in the stock market. For example, a change in the S&P 500 from 1000-2000 is different from a change from 8000-9000; using S&P returns corrects for this problem by looking at percentage change in the index over time.

I also look at the percentage change in GDP (on a quarterly basis) in my model. Under the assumption that if I am trying to predict the percentage change in S&P Returns, it makes sense to match the response variable with a percentage change predictor variable. Looking at percentage change for the predictor variables makes sense only with the GDP variable. It does not make sense to look at a percentage change in interest rates, since investors are concerned more with the level of the rates, rather than changes in them when making investment decisions. Furthermore looking at the change in corporate profits/ GDP also does not make as much sense as looking at the level of corporate profits relative to GDP in the economy; I therefore look at the level of this variable as well, rather than the percentage change.

The regressions presented later in my paper confirm the difficulty inherent in predicting changes in the stock market. As expected, the results of these regressions are quite weak.

Notes regarding my data:

• S&P Return was calculated as (S&P Index / S&P Index lagged one quarter) – 1

• GDP Change was calculated as (GDP / GDP lagged one quarter) - 1

• Real interest rates were calculated as: Nominal Rate – Inflation Rate

• The data source for the long-term t-bill rate used in my analysis, , makes the following adjustment to the data. “We use the Treasury Constant Maturity 20-Year T-Bond Yield until February 1977 and the Treasury Constant Maturity 30-Year T-Bond Yield to the present. The yield curve was flat at the 20-30 year maturity at the splice point.”[2]

• Data were tracked by quarter: January, April, July and October of each year 1978-1998.

• I was not able to find corporate profit data for 1997 or 1998; I applied the 1996 third and fourth quarter corporate profits/GDP result (which were the same) to all periods in 1997 and 1998. This is not an ideal solution to this problem; a more sophisticated analysis would more accurately correct for such problems.

• GDP data were not available for the most recent quarter, 10/2/98.

Data sources:

•

• National Income and Products Accountants NIPA

• bos.business.uab.edu

• ntu.edu.sg/library/statdata.htm

• stats.

Looking at my data, I observe the following trends as well as one outlier. Below are scatter plots showing S&P Index on the y-axes and each the 3 month t-bill, 20/30 year t-bill, GDP, change in GDP and corporate profits on the x-axes:

Judging from these scatter plots, there does not seem to be a strong relationship between any one predictor variable and the S&P Return. It is clear that one outlier exists. I have looked at the regressions both with and without this outlier, which occurred in the quarter 1/8/88.

Histograms of these same variables look as follows:

Long left tails in the t-bill histograms can be observed. GDP change is more normally distributed than GDP level. Corporate profits/ GDP is also fairly normally distributed.

Regressions

I looked at the regression of S&P Return versus each of the predictor variables. First running the regression of S&P Return against 3-month t-bill, GDP and corporate profits/GDP without removing the outlier revealed the following results for the residuals versus the order of the data, residuals versus the fitted values and the normal probability plot of the residuals.

Error/Residual Assumptions[3]

The graph of residuals versus the order of the data shows that there is no apparent relationship among the residuals; thus I can conclude that the residuals are not correlated. The residuals versus fits graph shows no apparent uniformity among the standard deviations of residuals and therefore heteroscedasticity is not a problem with these data. The third chart of normal probability of the residuals reveals that the error terms in the regression are fairly normally distributed.

The regression below looks at the effect of the same independent variables on S&P Return. I ran the regression first without removing the outlier.

Regression Analysis (1) – S&P Return versus Predictors

The regression equation is

S&P Return = 0.0752 - 0.00494 real 3 mo t-bill +0.000003 GDP

- 1.02 corp profit/gdp

83 cases used 1 cases contain missing values[4]

Predictor Coef StDev T P VIF

Constant 0.07516 0.05864 1.28 0.204

real 3 m -0.004935 0.005083 -0.97 0.335 1.6

GDP 0.00000278 0.00000464 0.60 0.550 1.0

corp pro -1.024 1.031 -0.99 0.324 1.6

S = 0.07747 R-Sq = 1.8% R-Sq(adj) = 0.0%

Analysis of Variance

Source DF SS MS F P

Regression 3 0.008775 0.002925 0.49 0.692

Residual Error 79 0.474129 0.006002

Total 82 0.482904

Source DF Seq SS

real 3 m 1 0.001081

GDP 1 0.001772

corp pro 1 0.005923

Unusual Observations

Obs real 3 m S&P Retu Fit StDev Fit Residual St Resid

11 -5.23 0.14988 0.05294 0.03512 0.09694 1.40 X

21 4.21 0.19029 0.02905 0.01541 0.16125 2.12R

38 1.79 0.21895 0.04237 0.01407 0.17658 2.32R

41 1.73 -0.25809 0.03820 0.01060 -0.29629 -3.86R

52 0.76 -0.13091 0.04552 0.01423 -0.17643 -2.32R

79 2.88 0.20982 0.03024 0.01832 0.17958 2.39R

84 2.77 -0.12545 0.03193 0.01964 -0.15738 -2.10R

R denotes an observation with a large standardized residual

X denotes an observation whose X value gives it large influence.

This regression revealed extremely low R-squared results which tells me that very little variability is accounted for by the predictor variables. The low t-statistics and high p-values observed for each of the predictor variables indicate that none of the predictor variables has much predictive value in this regression. The overall F statistic was also low indicating a weak regression. Variance Inflation Factors (VIFs) were all low indicating no collinearity among the data; however, because the predictors are all weak this isn’t really relevant. These poor results would be expected for my model, given the hypothesis posed.

Regression Model Removing Outliers

Rerunning the regression without the one outlier shows the following regarding error and residual assumptions:

Error/Residual Assumptions

These results from these graphs regarding error and residual assumptions are similar to the results described for the previous set of graphs. The only change is that the outlier is no longer evident.

Regression Analysis (1b) – S&P Return versus 3 month t-bill, GDP, corporate profits/GDP, removing the one Outlier

The regression equation is

S&P Return = 0.0966 - 0.00627 real 3 mo t-bill +0.000003 GDP

- 1.33 corp profit/gdp

82 cases used 1 cases contain missing values

Predictor Coef StDev T P VIF

Constant 0.09657 0.05340 1.81 0.074

real 3 m -0.006270 0.004619 -1.36 0.179 1.6

GDP 0.00000258 0.00000420 0.61 0.541 1.0

corp pro -1.3274 0.9372 -1.42 0.161 1.6

S = 0.07023 R-Sq = 3.3% R-Sq(adj) = 0.0%

Analysis of Variance

Source DF SS MS F P

Regression 3 0.012973 0.004324 0.88 0.457

Residual Error 78 0.384665 0.004932

Total 81 0.397639

Source DF Seq SS

real 3 m 1 0.001675

GDP 1 0.001406

corp pro 1 0.009892

Unusual Observations

Obs real 3 m S&P Retu Fit StDev Fit Residual St Resid

11 -5.23 0.14988 0.06425 0.03194 0.08563 1.37 X

21 4.21 0.19029 0.03388 0.01401 0.15641 2.27R

38 1.79 0.21895 0.04953 0.01286 0.16942 2.45R

51 0.76 -0.13091 0.05231 0.01300 -0.18322 -2.65R

78 2.88 0.20982 0.03032 0.01661 0.17950 2.63R

83 2.77 -0.12545 0.03206 0.01780 -0.15752 -2.32R

R denotes an observation with a large standardized residual

X denotes an observation whose X value gives it large influence.

The results of the regression are slightly stronger, but still quite weak (low p-values and high t-statistics for each independent variable, a low f-statistic, low r-squared and adjusted r-squared), but the outlier has been removed for completeness. The r-squared has increased from 1.8 to 3.3 after removing the outlier, but is still very low. The adjusted r-squared is 0.0%. The F-statistic is now .88, which is still low as well. The VIFs of all variables is low indicating that collinearity is not a problem, which is still not relevant given the fact that my variables are not predictive anyway. Based on these results, I still do not have sufficient evidence to reject the null hypothesis.

S&P Return versus 3 Month T-bill and Corporate Profits (levels) and Percentage Change in GDP

Running the regression of S&P Return against the percentage change in GDP (instead of the level of GDP as I had used before), 3-month interest rate and corporate profit/GDP, I would expect to see an improved result. It is intuitive to look at the quarterly change in GDP as a predictor variable when trying to predict the effect on the percentage change in the S&P. The t-statistic in this regression for GDP is 1.04, which is significant at a .30 level (i.e., there is a 30% probability that this is due to chance). While this t-statistic is not very high, it is interesting that the change in GDP has some impact on the S&P Return, particularly in light of how hard it is to predict change in S&P return.

Regression Analysis (2): S&P Return versus Percentage Change in GDP, 3-month t-bill and corporate profits/GDP

The regression equation is

S&P Return = 0.108 + 0.864 new gdp change - 0.00630 real 3 mo t-bill

- 1.60 corp profit/gdp

82 cases used 1 cases contain missing values

Predictor Coef StDev T P VIF

Constant 0.10760 0.04984 2.16 0.034

new gdp 0.8637 0.8334 1.04 0.303 1.1

real 3 m -0.006298 0.004590 -1.37 0.174 1.6

corp pro -1.5977 0.9764 -1.64 0.106 1.8

S = 0.06991 R-Sq = 4.1% R-Sq(adj) = 0.4%

Analysis of Variance

Source DF SS MS F P

Regression 3 0.016366 0.005455 1.12 0.348

Residual Error 78 0.381272 0.004888

Total 81 0.397639

Source DF Seq SS

new gdp 1 0.002148

real 3 m 1 0.001129

corp pro 1 0.013089

Unusual Observations

Obs new gdp S&P Retu Fit StDev Fit Residual St Resid

2 0.0587 -0.01583 0.05066 0.03412 -0.06649 -1.09 X

11 0.0235 0.14988 0.07385 0.03117 0.07603 1.21 X

12 0.0461 0.10106 0.07006 0.02731 0.03100 0.48 X

21 0.0188 0.19029 0.04318 0.01239 0.14711 2.14R

38 0.0174 0.21895 0.05381 0.01310 0.16513 2.40R

51 -0.0001 -0.13091 0.03725 0.01777 -0.16816 -2.49R

78 0.0133 0.20982 0.01758 0.01262 0.19223 2.80R

81 0.0067 0.15144 0.00797 0.01723 0.14347 2.12R

83 0.0086 -0.12545 0.01417 0.01445 -0.13962 -2.04R

R denotes an observation with a large standardized residual

X denotes an observation whose X value gives it large influence.

These results are slightly better for the GDP variable, i.e., it seems to make more sense to look at GDP change rather than GDP level, however, the variable is still not significant in the regression equation. The same can be said about the corporate profit variable – it is slightly higher in terms of its t-statistic and slightly lower in terms of its p-value, yet it is still not significant as a predictive variable. The overall F-statistic is still low at 1.12 as is the r-squared at 4.1% and the adjusted r-squared at 0.4%.

Copied below are the graphs showing residuals versus the order of the data, residuals versus the fitted values and the normal probability plot of the residuals for this regression. The residuals versus the order of the data show no relationship among the residuals; thus the residuals are not correlated. The graph of residuals versus the fitted values shows that the residuals are roughly normally distributed. The normal probability plot of the residuals shows that the errors are roughly normally distributed.

Regression Model with Time Variable Included

In an effort to observe whether the data followed a strong time trend, I ran the regression of S&P Returns versus the same predictor variables and adding a time variable called the “time dummy”.

Regression Analysis (2b): S&P Return vs. Percentage Change in GDP, 3 month t-bill, corporate profit/GDP and time variable

Regression Analysis

The regression equation is

S&P Return = 0.0874 + 1.51 new gdp change - 0.00688 real 3 mo t-bill

- 1.83 corp profit/gdp +0.000489 time dummy

82 cases used 1 cases contain missing values

Predictor Coef StDev T P VIF

Constant 0.08743 0.05191 1.68 0.096

new gdp 1.5070 0.9621 1.57 0.121 1.5

real 3 m -0.006884 0.004590 -1.50 0.138 1.6

corp pro -1.8288 0.9874 -1.85 0.068 1.8

time dum 0.0004892 0.0003707 1.32 0.191 1.4

S = 0.06959 R-Sq = 6.2% R-Sq(adj) = 1.4%

Analysis of Variance

Source DF SS MS F P

Regression 4 0.024799 0.006200 1.28 0.285

Residual Error 77 0.372840 0.004842

Total 81 0.397639

Source DF Seq SS

new gdp 1 0.002148

real 3 m 1 0.001129

corp pro 1 0.013089

time dum 1 0.008433

Unusual Observations

Obs new gdp S&P Retu Fit StDev Fit Residual St Resid

2 0.0587 -0.01583 0.05360 0.03403 -0.06943 -1.14 X

11 0.0235 0.14988 0.06468 0.03179 0.08520 1.38 X

21 0.0188 0.19029 0.03506 0.01378 0.15523 2.28R

38 0.0174 0.21895 0.05405 0.01304 0.16490 2.41R

51 -0.0001 -0.13091 0.03254 0.01804 -0.16345 -2.43R

78 0.0133 0.20982 0.03088 0.01610 0.17894 2.64R

83 0.0086 -0.12545 0.02694 0.01734 -0.15239 -2.26R

R denotes an observation with a large standardized residual

X denotes an observation whose X value gives it large influence.

The t-statistic and p-value for the time variable are 1.32 and .191, respectively, in this regression; based on these results, there does not appear to be a significant time trend with these data. Because adding the time variable does not significantly improve my model, I will not use it in my final the regression.

I feel that the regression without the time variable is the most evocative in terms of answering the question of the impact interest rates, GDP and corporate profits/GDP have on stock market returns. Therefore my focus will be on the following regression:

S&P Return versus 3-month t-bill, percentage change in GDP, and corporate profits/GDP (with the one outlier removed).

Regression Diagnostics

The diagnostics for this regression (i.e., S&P Return versus 3-month t-bill, percentage change in GDP, and corporate profits/GDP) reveal that the highest Cook’s distance is .057, which is not high enough to be concerned about; therefore, I can conclude that there are no influence points in these data.

With three predictors and 82 data cases, the relevant leverage factor is 2.5*(3+1) / 82 = 0.122. At this level, there are no cases with high leverage.

Looking again at the standard residuals, I see that there is one slight outlier which has a standard residual value of 2.579,which is slightly greater than 2.5. Because this standard residual is not too far above 2.5, I am not concerned about it.

Conclusion

The equation for my final model, which looks at the regression of the S&P Return against 3-month t-bill, percentage change in GDP, and corporate profits/GDP, after removing the one outlier is:

S&P Return = 0.108 + 0.864 new gdp change - 0.00630 real 3 mo t-bill

- 1.60 corp profit/gdp

Because the regression is not significant overall in terms of predictive quality, I cannot draw any conclusions regarding the variables. The intercept of 0.108 corresponds to the expected S&P Return when all independent variables are zero. This does not make any sense in this analysis, so I will ignore the intercept. If they were significant, the Beta coefficients would say that i) for every one percentage point increase in the GDP, the S&P 500 Return increases by 0.864 points; ii) for every one percentage point increase in three month t-bill rate, the S&P 500 Return decreases by 0.0063 points; and iii) for every one percentage point increase in corporate profits as a percent of GDP, the S&P 500 Return decreases by 1.6 points. However, because the model is not significant in terms of its predictive quality, I cannot draw any of these conclusions. The model does not show high p-values or t-statistics for any of the predictor variables – an expected result given the question being asked (the effect of these variables in predicting changes in the stock market). I am simply seeing random noise in this regression and therefore the signs of the coefficients don’t mean anything.

In summary, my regression has confirmed the fact that “stock prices seem to follow a random walk with no discernable predictable patterns that investors can exploit”.[5] As expected, it was not likely that I would see significant p-values and t-statistics for my independent variables or overall f-statistic, r-squared or adjusted r-squared, given the hypothesis I was testing. Therefore, I must to accept the null hypothesis that interest rates, (change in) GDP, and corporate profits/GDP have no effect on the stock market return. I could not get enough evidence against the null hypothesis in my regression models to reject the null.

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[1] It was necessary to look at corporate profits as a percent of GDP to see the true profits results independent of GDP.

[2]

[3] These graphs are presented first for the entire data set -- without removing the outlier.

[4] In each case, this missing value is GDP in the most recent quarter.

[5] Bodie, Kane, Marcus, Essentials of Investments, p.253.

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