Sample Project
Dr. Jantzen
ECO 310
Spring 2010
Term Project
An Econometric Analysis of the US Consumption
INTRODUCTION
The US economy is consumer driven. Personal Consumption Expenditures (PCE)
accounts for 70% of the United States’ Gross Domestic Product (GDP). In other words, more
than ⅔ of all goods and services produced each year in the US are represented by levels of
consumption. Therefore, the United States economy relies very heavily on its consumers. After
the economic meltdown that began in 2007, most economic measures focused on the American
consumer. Today, more than ever, personal consumption expenditures are vital to attaining a
sustainable economic recovery in America.
From an economic standpoint, personal consumption expenditures depend on a series
of factors. Using multiple regression analysis, this paper will examine the effects of three
explanatory variables: Disposable Personal Income (DSPI), Inflation Rate (CPI) and Bank Prime
Loan Rate (MPRIME) on Consumption Expenditure (PCE) in the United States, from January
1992 to January 2001. PCE and DSPI are measured in billions of dollars, while CPI and
MPRIME are measured as percentages.
LITERATURE REVIEW
In 2007, Michael Curran conducted a similar econometric study in order to examine the
relationship between Consumption Expenditures and two explainers, Income and Interest
Rates. Michael’s study sought to investigate the Keynes’ Consumption Function theory,
focusing primarily on the effects of real income per capita and nominal interest rates (adjusted
for inflation) on real consumption expenditure per capita in the United States.1 In his analysis,
Michael used data from the first quarter in 1949 through the third quarter in 2006.
2
First, Michael Curran plotted personal consumption expenditure (PCE) against personal
disposable income (PDI) to show the close relationship between the two variables and the
upward trend- more disposable income indicates higher consumption expenditure. Yet, when
1, 2
Curran, M. (2007). "Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in
Postwar America" Student Economic Review, Volume 21:59-72.
Michael plotted PCE against the bank prime loan rate (PRIME), it became very hard to identify a
clear correlation between the two variables.3 Lastly, when Michael plotted PDI against PRIME,
the graph depicted a scattered behavior. Thus, he concluded that there was no multicollinearity
between his explanatory variables.4 Below are the graphical illustrations from Michael Curran’s
analysis:
PCE against PDI
PCE against PRIME
3, 4
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
Changes in PDI against changes in PRIME
Michael’s econometric study used time-series data. After employing his regression
models and examining his estimated coefficients (β), Michael observed that his three variables
PCE, PDI and PRIME were non-stationary.5 In other words, the results showed that there was
no observable trend in his time series variables – no constant mean or variance was detected.
According to Michael’s findings, the averages for PCE and PDI “respectively rise over time, and
the variation in PRIME also changes over time.6
After running a Durbin-Watson test, Michael obtained a D-W sample value of .245. He
noted that because of OLS (Ordinary Least Squares), the estimated results were bogus
(biased).7 Since Michael was not able to reject the null hypothesis, he concluded that the
correlation was serial, and therefore, OLS results should not be used. In order account for the
misleading results, Michael re-estimated his model using a new equation:
ΔPCEt = β0 + β1 ΔPDIt + βt ΔPRIMEt + ut^5
where:
ΔPCE = quarterly change in real personal consumption expenditure per person employed.
ΔPDI = quarterly change in real personal disposable income per person employed.
ΔPRIME = quarterly change in bank prime loan rate.
4, 5
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
5,6
7
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
u = residual.8
After running descriptive statistics in his estimated model, Michael came across the
following summary, followed by the new regression results:
Variable
PCE (US$)
PDI (US$)
PRIME (%)
Maximum
31,895.80
33,483.40
20.3233
Minimum
12,654.10
13,619.20
2
Mean
22,166.20
24,551.40
7.1076
5,101.00
3.4355
Std. Deviation4,904.80
Avg. Growth 12 0.004079 0.003881
0.015892
Regression Results:
Regressor
Coefficient
Standard Error
T-Ratio
[Prob]
CONSTANT 52.9116
11.7046
4.5206
[.000]
ΔPDI
0.38058
0.043485
8.7521
[.000]
ΔPRIME
-34.5743
10.8581
3.1842
[.002]
Relevant Statistics:
Statistic Value
R-Squared
0.30257
R-Bar-Squared
0.29643
F-Statistic F (2,227): 49.2406 [.000]
DW-statistic
2.3422
The R-squared value of .30257 indicated that only 30% of all variations in PCE were tied
to Personal Disposable Income and Interest Rates. Michael concluded that there was “sufficient
evidence of closeness to fit”9. However, his model represents low fitness, as 70% of the
variations in his dependent variable are not “explained” by his explainers. Furthermore, Michael
conducted T-Tests on his population coefficients and found that the regression coefficients β0,
β1 and β2 were all different than zero, indicating that all of his explainers have an effect on
PCE, the dependent variable.
9
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
According to Michael, the new Durbin-Watson test displayed evidence of a negative
autocorrelation10. He further used the Breusch-Godfrey test and concluded that serial correlation
was present in his econometric model.11 Towards the end of his study, Michael used a 95%
Confidence Interval for all of the population coefficients in order to determine the range under
which the real population coefficient would lie. In his conclusion, Michael Curran stated that
Keynes would have been proud of his linear regression results, provided that he agreed with his
interpretation.12
DATA AND METHODOLOGY
In order to examine the effects of disposable personal income, inflation rate and bank
prime loan rate on personal consumption expenditures in the US, a database containing
monthly data from 1992 to 2003 was used. Since this is a time series analysis, the following
estimated model was used:
Yt= β0 + β1DSPIt + β2CPIt + β3MPRIMEt + Et
where:
Dependent variable= Y - Personal Consumption Expenditure (PCE)
Explainers= DSPI (Disposable Personal Income), CPI (Inflation Rate) and MPRIME (Bank
Prime Loan Rate)
Residual | Error term= Et
β0 = Constant – expected value for Y if explainers are equal to zero.
β1, β2, β3= population regression coefficients- shows how many units the dependent variable
will change if explainer changes by one unit.
The model suggests that there is a relationship between the dependent variable and the
explainers. It is believed that disposable income directly impacts consumer spending. As
evidenced by the recent recession, during times of economic uncertainty, consumer spending
significantly declines. It is also anticipated that consumption expenditure is sensitive to inflation
because higher inflation reduces the consumer’s buying power. With regard to interest rate, the
Prime Rate is also expected to impact consumer spending as the rate is directly tied to lending
and access to credit.
10, 11, 12
"Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar America"
This analysis will further assume the presence of statistical problems such as
Multicollinearity and Autocorrelation, and make corrections where necessary in order to improve
the statistical results.
EMPIRICAL RESULTS
Obtaining descriptive statistics is among the first steps in a regression analysis, because it
summarizes the data in an easy and understandable way.
Descriptive Statistics:
Variable
Mean
Std.Dev.
Minimum
Maximum
Cases
===============================================================================
------------------------------------------------------------------------------All observations in current sample
------------------------------------------------------------------------------PCE
5667.44060
1016.68993
4108.50000
7477.50000
133
DSPI
6127.94361
1006.34991
4633.30000
8024.30000
133
CPI
2.56917293
.650894542
1.10000000
3.80000000
133
MPRIME
7.44037594
1.44765244
4.25000000
9.50000000
133
The above table depicts the averages, standard deviations, minimums and maximums,
for a 133-month period in the United States. According to the table, the average for
consumption expenditure is 5667.4 billions of dollars, 6127.9 billions of dollars for personal
disposable income, 2.57% (annualized) for CPI inflation, and 7.44% for bank prime rate. The
standard deviations show how much the numbers differ from the average. The minimum and
maximum amounts indicate the smallest and highest value (in billions of dollars or percentages)
for each variable.
Since the explainers DSPI, CPI and MPRIME do not have strong correlation coefficients
(≥|.9|), there is no problem of multicollinearity with this analysis. See correlation results below:
Correlation Matrix for Listed Variables:
PCE
DSPI
CPI
PCE
1.00000
.99812
-.30542
DSPI
.99812
1.00000
-.31543
CPI
-.30542
-.31543
1.00000
MPRIME
-.06624
-.08853
.28436
MPRIME
-.06624
-.08853
.28436
1.00000
Ordinary Least Squares Regression Results:
+-----------------------------------------------------------------------+
| Ordinary
least squares regression
Weighting variable = none
|
| Dep. var. = PCE
Mean=
5667.440602
, S.D.=
1016.689929
|
| Model size: Observations =
133, Parameters =
4, Deg.Fr.=
129 |
| Residuals: Sum of squares= 444215.9562
, Std.Dev.=
58.68164 |
| Fit:
R-squared= .996744, Adjusted R-squared =
.99667 |
| Model test: F[ 3,
129] =13164.64,
Prob value =
.00000 |
| Diagnostic: Log-L =
-728.2810, Restricted(b=0) Log-L =
-1109.1498 |
|
LogAmemiyaPrCrt.=
8.174, Akaike Info. Crt.=
11.012 |
| Autocorrel: Durbin-Watson Statistic =
.79803,
Rho =
.60099 |
+-----------------------------------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
-660.3852885
49.462718 -13.351
.0000
DSPI
1.011677878
.53484069E-02 189.155
.0000
6127.9436
CPI
6.938780130
8.5913921
.808
.4208
2.5691729
PRIME
14.85059038
3.6801083
4.035
.0001
7.4403759 13
13
Note: E+nn or E-nn means multiply by 10 to + or -nn power.
After plotting the residuals against the constant variable and observing the shape of the
graph, it became evident that the errors could be serially correlated. Serially correlated errors
imply that the errors follow a certain pattern. For this analysis it means that if last month had a
negative correlation, this month will most likely be negative too (this period depends on what
happened last period). From the graph, we clearly see the pattern: all negative, then all positive,
then all negative again… Assuming that the errors are serially correlated, an adjusted model
was used:
Yt= β0 + β1DSPIt + β2CPIt + β3MPRIMEt + Et
where
Et= ρEt-1 + Vt
% of last month’s error terms + random factor
With serial correlation, T-stats are usually too big, because standard errors are too
small. Serial correlation
Durbin-Watson Test:
In order to test whether the OLS regression results suffer from serial correlation, the
Durbin-Watson test was used:
Hypothesis: Null HO:
Alternative HA:
ρ=0
ρ >0
Sample D-W Statistics:
.798
Critical D-W Statistics:
dL= 1.61
dU= 1.74 @ 5% significance level
Decision: Since sample number is < dL, the null is rejected and we
can be 95% confident that there is a positive correlation between
last month and this month’s term. Therefore, we have serial
correlation and cannot rely on or use OLS regression results.
When the terms are serially correlated, there are consequences associated with using
OLS estimated results. First, the estimated errors of the coefficients (β) will be biased downward
(too small) causing the sample T’s to be too big. Hence, we are inclined to reject the null
hypothesis more than we should. Second, the formulas used for finding the OLS coefficients,
while unbiased, are inefficient. In other words, there is a lot of dispersion around the true betaβ.
Aiming to correct the serial correlation, the Prais-Winsten correction was employed. The
Prais-Winsted correction gets rid of serial correlation and generates betas that are unbiased and
have the correct error correlation- new sample T’s and standard errors are calculated after the
test is ran.
+---------+--------------+----------------+--------+---------+----------+14
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
-570.4907167
94.985620
-6.006
.0000
DSPI
1.002101497
.10593107E-01
94.599
.0000
6127.9436
CPI
5.844973323
13.445927
.435
.6638
2.5691729
MPRIME
10.84200799
7.1723606
1.512
.1306
7.4403759
RHO
.6225894017
.68112120E-01
9.141
.0000
Finally, after comparing the Prais-Winsten corrected results to the OLS results, it was
concluded that the OLS standard errors and t-ratios are unreliable for this type of analysis. The
estimated coefficients were also significantly different between OLS and the corrected results.
Estimated Coefficients and Confidence Intervals:
Estimated coefficients show how many units the dependent variable will change if the
explainer changes by one unit. Based on the re-estimated regression with the correction for
serial correlation, one could be tempted to interpret the coefficients as the following:

14
If DSPI changes by 1 billion of dollars, PCE will change by 1 billion of dollars.
Note: E+nn or E-nn means multiply by 10 to + or -nn power.

If CPI changes by 1 percent (annualized), PCE will change by 5.84 billions of
dollars.

If MPRIME changes by 1 percent, PCE will change by 10.84 billions of dollars.
However, such interpretation can only be valid given that the explainers have a real
effect upon the dependent variable. Therefore, T-tests were conducted on the population
regressions coefficients to make sure that all of the explainers had an effect on the PCE. The
results were very surprising.
HO:
HA:
Sample T:
Critical T:
Decision:
β1= 0
β1≠ 0
β2= 0
β2≠ 0
β3= 0
β3≠ 0
94.6
0.44
1.5
1.98 @ 5% Significance Level
β1: Reject the null. 95% sure that the coefficient on DSPI is ≠ 0. DSPI has an effect on
PCE.
β2: Can't reject the null. There's not enough evidence that CPI affects PCE.
β3: Can't reject the null. There's not enough evidence that MPRIME affects
PCE.
As evidenced by the above T-tests results, only Disposable Personal Income was
found to have an effect on PCE. There was not enough evidence to state that the coefficients on
CPI and MPRIME were different than zero, and therefore, we cannot state with any confidence
that CPI and MPRIME have an effect on Personal Consumption Expenditure.
Moreover, the Confidence Interval (CI) results also supported the T-test findings.
The formula used for CI in this economic study was B ± (critical t x standard error of B) . The
90% Confidence Interval for the population B1 went from 0.98 to 1.02. Therefore, we are 90%
sure that the real population coefficient for B1 is between .98 and 1.02. More disposable income
means more consumption. On the contrary, the 90% Confidence Interval for the population B2
was between -16.49 and 28.08. Hence, we are 90% confident that the real population coefficient
is among the range obtained, and there is no evidence that CPI has an effect on PCE (could be
positive, could be negative or could be zero). Similarly, the 90% Confidence Interval for the
population B3 also indicated that there is no evidence that MPRIME has an effect on consumer
spending, since the range was -1.15 to 22.75.
Goodness of Fit:
The quality of the model can be assessed through the R-squared. The R-squared
measures the proportion of the variation in the dependent variable that is determined by
differences in the explainer’s number. The formula for computing the R-squared is:
1 – (Unexplained Variation/ Total Variation)
= .9966
Total Variation= (Standard Deviation of Y)^2 x (N-1) ; where N= sample size
(136445613.5)
Unexplained Variation= (e (t))^2 x (N- # coefficients)
(462853.29)
**The numbers in green are the answers to the mathematical computations.
There no major difference between the OLS R-squared and the R-squared for the
corrected results. But the above R-squared value of .997 indicates that this model has high
fitness, as 99% of the variations in PCE are tied to differences in DSPI, CPI, and MPRIME.
However, since there’s no evidence that CPI and MPRIME affect PCE, most of the variations in
PCE may be attributed to changes in personal disposable income alone.
Standardized Coefficients:
Standardized coefficients show how many standard deviations the dependent variable
will change if the explainer changes by one standard deviation. In short, standardized
coefficients are used in picking which explainer is the most important; the closer to 1, the
stronger the relationship between the dependent variable and explainer.
Formula:
Β* = β x (standard deviation of explainer/ standard deviation of dependent variable)
After making calculations, the following results were collected for the estimated
coefficients:
Β1= .98 Consumption expenditure will change .98 standard deviations if personal income
changes by one standard deviation.
B2= .0037 Consumption expenditure will change .0037 standard deviations if CPI inflation
changes by one standard deviation.
B3= .015 Consumption expenditure will change .015 standard deviations if the bank prime rate
changes by one standard deviation.
Thus, disposable personal income (DSPI) has the greatest effect on personal
consumption expenditures (PCE). Based on the standardized coefficients results, CPI and
MPRIME seem to have a relatively small effect on the dependent variable.
Specification Bias:
Leaving out an important explanatory variable can lead to biased results. In this analysis,
the true model is Yt= β0 + β1DSPIt + β2CPIt + β3MPRIMEt + Et. But let’s suppose that we
decided to isolate the DSPI variable after the standardized coefficient results. The equation then
becomes Yt= β0 + β2CPIt + β3MPRIMEt + Et
The sign of the bias on any explainer’s coefficient = sign
of the omitted explainer’s coefficient x partial correlation
between omitted and examined explainers.
Therefore,
B2 = + * - = negative
Any coefficient we get for B2 will be too small.
B3= + * - = negative
Any coefficient we get for B3 will be too small.
Specification errors emphasize the unreliability of OLS estimated results. The bias can
be observed in two forms: First, if we omit one or more relevant explainers, and secondly, if we
include one or more irrelevant explainers. We should not be worried about including variables
that do not belong. However, it is imperative that we do not leave out important explanatory
variables like disposable personal income.
CONCLUSION
After a thorough econometric analysis on the effects of DSPI, CPI and MPRIME on
Personal Consumption Expenditures, a few important implications were drawn. First and
foremost, one cannot rely on Ordinary Least Squares regression results, because it can be
deceptive and bias. Furthermore, time series data can often result in serially correlated error
terms, also known as Autocorrelation. Accordingly, it is important to identify the pattern and
make the necessary corrections. Based on the R-squared value obtained through the corrected
regression results, one can argue that this is a high quality model. Nevertheless, the T-Test
results on the population coefficients and the Confidence Intervals findings indicate a lack of
evidence to explain the effect of two out of three explanatory variables (CPI and MPRIME) on
PCE. It is possible that collecting more data could improve the results and the model itself.
Overall, it became obvious that disposable income has the biggest effect on US consumption
expenditure, with an almost perfect (=one) correlation.
Reference:
Curran, M. (2007). "Keynes Re-interpreted - An Econometric Investigation of Keynes' Consumption Function Theory in Postwar
America" Student Economic Review, Volume 21:59-72.