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 profits1 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
1
It was necessary to look at corporate profits as a percent of GDP to see the true profits results independent
of GDP.
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
2

Real interest rates were calculated as: Nominal Rate – Inflation Rate

The data source for the long-term t-bill rate used in my analysis, TradeTools.com,
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 30Year 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 19781998.

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:





www.tradetools.com
National Income and Products Accountants www.NIPA
www.bos.business.uab.edu
www.ntu.edu.sg/library/statdata.htm
www.stats.bls.gov
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 tbill, GDP, change in GDP and corporate profits on the x-axes:
0.2
S&P Return
0.1
0.0
-0.1
-0.2
-0.3
-5
0
5
real 3 mo t-bill
2
www.tradetools.com
3
0.2
S&P Return
0.1
0.0
-0.1
-0.2
-0.3
0
5
10
real 20/30 tbill
0.2
S&P Return
0.1
0.0
-0.1
-0.2
-0.3
2000
3000
4000
5000
6000
7000
8000
9000
GDP
4
0.2
S&P Return
0.1
0.0
-0.1
-0.2
-0.3
0
50
100
150
GDP change
0.2
S&P Return
0.1
0.0
-0.1
-0.2
-0.3
0.03
0.04
0.05
0.06
0.07
corp profit/gdp
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:
5
Frequency
20
10
0
-5
0
5
real 3 mo t-bill
9
8
Frequency
7
6
5
4
3
2
1
0
2000
3000
4000
5000
6000
7000
8000
9000
GDP
Frequency
20
10
0
0
5
10
real 20/30 tbill
6
Frequency
15
10
5
0
0
50
100
150
GDP change
Frequency
20
10
0
0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065 0.070
corp profit/gdp
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.
7
Error/Residual Assumptions3
Residuals Versus the Order of the Data
(response is S&P Retu)
3
Standardized Residual
2
1
0
-1
-2
-3
-4
10
20
30
40
50
60
70
80
Observation Order
Residuals Versus the Fitted Values
(response is S&P Retu)
3
Standardized Residual
2
1
0
-1
-2
-3
-4
0.01
0.02
0.03
0.04
0.05
Fitted Value
3
These graphs are presented first for the entire data set -- without removing the outlier.
8
Normal Probability Plot of the Residuals
(response is S&P Retu)
2.5
2.0
Normal Score
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-4
-3
-2
-1
0
1
2
3
Standardized Residual
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 values4
Predictor
Constant
real 3 m
GDP
corp pro
Coef
0.07516
-0.004935
0.00000278
-1.024
S = 0.07747
StDev
0.05864
0.005083
0.00000464
1.031
R-Sq = 1.8%
T
1.28
-0.97
0.60
-0.99
P
0.204
0.335
0.550
0.324
VIF
1.6
1.0
1.6
R-Sq(adj) = 0.0%
Analysis of Variance
Source
Regression
Residual Error
Total
4
DF
3
79
82
SS
0.008775
0.474129
0.482904
MS
0.002925
0.006002
In each case, this missing value is GDP in the most recent quarter.
9
F
0.49
P
0.692
Source
real 3 m
GDP
corp pro
DF
1
1
1
Seq SS
0.001081
0.001772
0.005923
Unusual Observations
Obs
real 3 m
S&P Retu
Resid
11
-5.23
0.14988
1.40 X
21
4.21
0.19029
2.12R
38
1.79
0.21895
2.32R
41
1.73
-0.25809
3.86R
52
0.76
-0.13091
2.32R
79
2.88
0.20982
2.39R
84
2.77
-0.12545
2.10R
Fit
StDev Fit
Residual
St
0.05294
0.03512
0.09694
0.02905
0.01541
0.16125
0.04237
0.01407
0.17658
0.03820
0.01060
-0.29629
-
0.04552
0.01423
-0.17643
-
0.03024
0.01832
0.17958
0.03193
0.01964
-0.15738
-
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 pvalues 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:
10
Error/Residual Assumptions
Residuals Versus the Order of the Data
(response is S&P Retu)
Standardized Residual
3
2
1
0
-1
-2
-3
10
20
30
40
50
60
70
80
Observation Order
Residuals Versus the Fitted Values
(response is S&P Retu)
Standardized Residual
3
2
1
0
-1
-2
-3
0.01
0.02
0.03
0.04
0.05
0.06
0.07
2
3
Fitted Value
Normal Probability Plot of the Residuals
(response is S&P Retu)
2.5
2.0
Normal Score
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3
-2
-1
0
1
Standardized Residual
11
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
Constant
real 3 m
GDP
corp pro
Coef
0.09657
-0.006270
0.00000258
-1.3274
S = 0.07023
StDev
0.05340
0.004619
0.00000420
0.9372
R-Sq = 3.3%
T
1.81
-1.36
0.61
-1.42
P
0.074
0.179
0.541
0.161
VIF
1.6
1.0
1.6
R-Sq(adj) = 0.0%
Analysis of Variance
Source
Regression
Residual Error
Total
Source
real 3 m
GDP
corp pro
DF
1
1
1
DF
3
78
81
SS
0.012973
0.384665
0.397639
MS
0.004324
0.004932
F
0.88
P
0.457
Seq SS
0.001675
0.001406
0.009892
Unusual Observations
Obs
real 3 m
S&P Retu
Resid
11
-5.23
0.14988
1.37 X
21
4.21
0.19029
2.27R
38
1.79
0.21895
2.45R
51
0.76
-0.13091
2.65R
78
2.88
0.20982
2.63R
83
2.77
-0.12545
2.32R
Fit
StDev Fit
Residual
0.06425
0.03194
0.08563
0.03388
0.01401
0.15641
0.04953
0.01286
0.16942
0.05231
0.01300
-0.18322
0.03032
0.01661
0.17950
0.03206
0.01780
-0.15752
St
-
-
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
12
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, 3month 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
Constant
new gdp
real 3 m
corp pro
Coef
0.10760
0.8637
-0.006298
-1.5977
S = 0.06991
StDev
0.04984
0.8334
0.004590
0.9764
R-Sq = 4.1%
T
2.16
1.04
-1.37
-1.64
P
0.034
0.303
0.174
0.106
VIF
1.1
1.6
1.8
R-Sq(adj) = 0.4%
Analysis of Variance
Source
Regression
Residual Error
Total
Source
new gdp
real 3 m
corp pro
DF
1
1
1
DF
3
78
81
SS
0.016366
0.381272
0.397639
MS
0.005455
0.004888
F
1.12
P
0.348
Seq SS
0.002148
0.001129
0.013089
Unusual Observations
Obs
new gdp
S&P Retu
Resid
Fit
13
StDev Fit
Residual
St
2
0.0587
1.09 X
11
0.0235
1.21 X
12
0.0461
0.48 X
21
0.0188
2.14R
38
0.0174
2.40R
51
-0.0001
2.49R
78
0.0133
2.80R
81
0.0067
2.12R
83
0.0086
2.04R
-0.01583
0.05066
0.03412
-0.06649
0.14988
0.07385
0.03117
0.07603
0.10106
0.07006
0.02731
0.03100
0.19029
0.04318
0.01239
0.14711
0.21895
0.05381
0.01310
0.16513
-0.13091
0.03725
0.01777
-0.16816
0.20982
0.01758
0.01262
0.19223
0.15144
0.00797
0.01723
0.14347
-0.12545
0.01417
0.01445
-0.13962
-
-
-
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.
14
Residuals Versus the Order of the Data
(response is S&P Retu)
Standardized Residual
3
2
1
0
-1
-2
-3
10
20
30
40
50
60
70
80
Observation Order
Residuals Versus the Fitted Values
(response is S&P Retu)
Standardized Residual
3
2
1
0
-1
-2
-3
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fitted Value
Normal Probability Plot of the Residuals
(response is S&P Retu)
2.5
2.0
Normal Score
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3
-2
-1
0
1
2
Standardized Residual
15
3
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
Constant
new gdp
real 3 m
corp pro
time dum
Coef
0.08743
1.5070
-0.006884
-1.8288
0.0004892
S = 0.06959
StDev
0.05191
0.9621
0.004590
0.9874
0.0003707
R-Sq = 6.2%
T
1.68
1.57
-1.50
-1.85
1.32
P
0.096
0.121
0.138
0.068
0.191
VIF
1.5
1.6
1.8
1.4
R-Sq(adj) = 1.4%
Analysis of Variance
Source
Regression
Residual Error
Total
Source
new gdp
real 3 m
corp pro
time dum
DF
1
1
1
1
DF
4
77
81
SS
0.024799
0.372840
0.397639
MS
0.006200
0.004842
F
1.28
P
0.285
Seq SS
0.002148
0.001129
0.013089
0.008433
Unusual Observations
Obs
new gdp
S&P Retu
Resid
2
0.0587
-0.01583
1.14 X
11
0.0235
0.14988
1.38 X
21
0.0188
0.19029
2.28R
38
0.0174
0.21895
2.41R
51
-0.0001
-0.13091
2.43R
Fit
StDev Fit
Residual
0.05360
0.03403
-0.06943
0.06468
0.03179
0.08520
0.03506
0.01378
0.15523
0.05405
0.01304
0.16490
0.03254
0.01804
-0.16345
16
St
-
-
78
2.64R
83
2.26R
0.0133
0.20982
0.03088
0.01610
0.17894
0.0086
-0.12545
0.02694
0.01734
-0.15239
-
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
17
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.
5
Bodie, Kane, Marcus, Essentials of Investments, p.253.
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