Paper 2 Regression Analysis

8/18/2019 Paper 2 Regression Analysis

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Regression Analysis:The relationship between real personal consumption expenditures per

capita and real disposable personal income per capita, Wealth,

Unemployment, Interest Rates, Oil Shocks, and the OPEC oil Embargo

Erik Robinson

Monday, March 03, 2015

Intermediate Macroeconomics

Professor Diana Fuguitt


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Introduction

In the pre modern era, when trade was miniscule compared to the economie of

today’s era; there was little to no middle class. Instead most of the consumption was

between the merchants and the Rich aristocrats of the day. However, today the middle

class is a large component of society and the consumers as a whole contribute to about

70% of the United States GDP. With a factor this large, it is important to understand

exactly what variables affect consumers’ consumption, and their relationship. If

acquired, the implications of such knowledge is powerful and would give economists

and market watchers a better understanding of how the world works and allow

economists to better predict how the market will respond to different events.

Perhaps in pursuit of such knowledge and understanding, Keynes did just that.

He hypothesized that there is a direct relationship between how much people consume

and their personal disposable income (income available for spending after taxes). In the

Keynes model the more money people make, the more they spend.

As practitioners, following in the footsteps of the great Adam Smith and John

Maynard Keynes, it is our duty to test such a theory and make sure it still holds. This

paper will perform regression analysis for the years 1962-1994 to seek to identify two

things. The first will be to distinguish if in fact there is a statistically significant

relationship between people’s real personal consumption expenditure and real

disposable income per capita, unemployment, %S&P, real interest rate, Opec Oil

embargo, and oil shocks. The second will be to measure the size of the relationship


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between peoples consumption for each additional dollar of disposable income and any

one percentage increase in the S&P, unemployment, real interest rate; as well as how an

oil shock or the Opec oil embargo effects consumption.

Model

The model for the consumption function is presented in table 1.

Constant ‘a’ or autonomous Cis the amount of per capita consumption when all

independent variables are zero. This means that when income is zero, Interest rates are

zero, %S&P is zero and there is no OPEC/Oil shocks constant ‘a’ is the amount of

consumption per capita people consume. Constant ‘a’ is expected to be greater > 0. The

reasoning is that people have to spend a certain amount of money to survive. If they

have no disposable income, they are still spending money either from borrowing or

spending their savings. Constant ‘a’ is also the y intercept.

Income or Coefficient ‘b’ is important to economists; the coefficient for income is

considered the marginal propensity to consume (MPC). Coefficient ‘b’ (MPC) is also

expected to be greater than 0 because in the Keyenes model it is expected that for each

additional dollar of disposable income received, an individual will choose so spend a

proportion of that dollar and save a proportion of that dollar. This is important to the

analysis of consumption because coefficient ‘b’ is the induced part of spending and

indicates the proportion an individual spends of one more dollar of real disposable

income. Alternatively, 1-MPC is considered the marginal propensity to save (MPS) or


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the proportion an individual saves of one more dollar of real disposable income. Table 1

presents the entire consumption model.

Interest Rate the coefficient for ‘R’ measures the relationship between

consumption and an increase in the real interest rate. ‘R’ is the rate individuals or

businesses in the economy are charged to borrow money from a financial institution.

For this reason, in the Keynes model, the coefficient for ‘R’ is expected to have a

negative sign indicating that for an increase in the real interest rate; per capita

consumption is expected to decrease by a certain amount.

Unemployment or coefficient ‘U’ has an inverse relationship to income. In the

Keynes model, as unemployment increases, it is expected that consumption will

decrease as more people are without work and have no income cash flow. For this

reason, the expected sign for coefficient ‘U’ is a negative sign.

Stock Portfolio Wealth or coefficient ‘f%S&P’ measures the relationship between

consumption and wealth. In the Keynes model; for any percentage point increase in the

stock market, people will have more income and consume more. For this reason,

coefficient ‘f%S&P’ is expected to have a positive sign.

OPEC oil embargo or coefficient ‘gOPEC’ measures the relationship between

consumption and the OPEC oil embargo in the years of 1973 and 1974. Professor

Hamilton at the University of California, San Diego reported the effects of the OPEC oil

embargo and oil shocks in his paper “Historical Oil Shocks” (2011). He reports that the

embargo and the shocks resulted in a shortage of oil. Resulting from the shortage,


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consumers were lining up at gas stations for hours, there was a general uncertainty

among consumers as to what was going to happen in the economy (Hamilton, 2011).

With this uncertainty about the future, it is expected that consumers reduced their

consumption or alternatively held on to their money. For this reason, the expected sign

for coefficient gOPEC is negative.

Oil Shock or coefficient SHOCK measures how consumption is affected by the oil

shocks 1979 and 1980. Resulting from the same logic described for the OPEC oil

embargo, the expected sign for coefficient SHOCK is a negative sign.

Table 1: Consumption Model

C= a + by +dR +eU + f%S&P + gOPEC + hSHOCK + є

C= real personal consumption expenditure; chained 2009 billions of dollars

Autonomous consumption ‘a’= consumption when all independent variables are

zero; Also the y intercept.

Induced consumption ‘b’= induced consumption; the proportion of consumption

that responds to a change in ‘y’. Also the MPC.

Income ‘y’ = Personal disposable income; chained 2009 billions of dollars.

dR = The real interest rate.

S&P Index ‘f%S&P’ = measures the effects of consumer wealth on real per capitapersonal consumption .

gOPEC = measures the effects of oil embargos on real per capita consumption.

hSHOCK = the effects of oil shocks on real per capita consumption.


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Hypotheses

Hypothesis testing is crucial in determining the statistical significance of various

components of a regression. There are 3 types of hypothesis testing, the F-test and the T-

test and Durbin Watson test. In all 3 tests, we set up two hypothesis statements; a null

and an alternative hypothesis (see Table 2). Establishing these two hypotheses is critical

in determining whether or not the model/coefficients are statistically significant from

zero or in the case of the Durbin Watson test, if the error terms are serially independent

or if they are not serially independent. In all tests, we either reject or do not reject the

null hypothesis. If the null hypothesis is rejected, it is appropriate to affirm the

alternative hypothesis.

The F-test indicates if the regression is statistically significant, if the null

hypothesis cannot be rejected; this would indicate that the coefficient of multiple

determination ( R2 ) is not significantly different from zero. However, if the conclusion is

to reject the null hypothesis, this would indicate that the R2 is statistically different from

zero. If the R2 is significantly different from zero, this indicates that a proportion of

variation of consumption is predicted by a variation of real per capita disposable

personal income, unemployment, real interest rate, %S&P, oil shocks, and Opec Oil

embargo . To either reject or not reject the null hypothesis, an F-calculated value (F-calc)


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and an F-critical value (F-crit) must be obtained and compared. The degree of certainty

we can either reject or not reject the null hypothesis is dependent at which level of

significance (a) we obtain our F-critical value. In our analysis, the F-crit is obtained at a

5% significance level.

The t-test assesses if each coefficient is significantly different from zero. If the

coefficient is found to be significantly different from zero, this indicates that the

particular coefficient being measured influences consumption. This test requires a

hypothesis for each coefficient. The null hypothesis states if we reject the null

hypothesis; the corresponding coefficient is statistically significant and we can affirm

the alternative hypothesis, but if we do not reject the null hypothesis the corresponding

coefficient is considered not significantly different from zero. Refer to Table 2.

The Durbin-Watson statistic test analyzes at a particular level of significance if

the error terms are serially independent or not serially independent. If the error terms

are serially independent; this means that the successive residuals vary in value in a

random (independent) manner, but if they are not serially independent, the successive

residuals vary in a systematic manner. The null hypothesis states that the error terms

are serially independent. The alternative hypothesis states that the error terms are not

serially independent. Unlike the F-test and t-test where we seek to reject the null, and

affirm the alternative. In the Durbin Watson test; we seek to not reject the null

hypothesis. To either reject or not reject the null hypothesis the d test statistic must be

compared to d critical limits found on a d-distribution table. If d < dL or d > 4-dL reject


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the null hypothesis and affirm at the .05 level of significance there is significant

evidence of serial correlation. If d > du but d is < (4-du) do not reject the null hypothesis;

there is not significant evidence of serial correlation at a .05 significance level.

Table 2: The Hypotheses

F-test: Ho: The regression is not statistically significant

Ha: The regression is statistically significant

T-test: Ho: coefficient = 0

Ha: coefficient ≠ 0

d-test H0: error terms are serially independent

Ha: error terms are not serially independent

Methodology

After developing the hypotheses, it is imperative to run a linear regression

analysis; so that we may be able to make predictions based on the data. The least

squares method estimates a “line of best fit” throughout the plotted data points. The

line of best fit, seeks to minimize the sum of the residuals (y-y’) squared. A residual is

the difference between the observed data (actual data) and the point predicted by the

regression. The further the observed data is away from the line, the greater the sum of

the residuals.


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Predictions are subject to error; for this reason, a standard error of the estimate

(SEest) is calculated for the model. The SEest measures the standard deviation of the

scatter of observed values of Y around the corresponding predicted value of Y’. Taking

into account the likely error, this regression analysis will report the 95% confidence

interval for each predicted value of consumption. The SEest was obtained by the

formula:

=Σ(Y Y′ )

2

Resulting from likely errors from the regression, it is appropriate to report Y’

values with a certain level of confidence. This means if we take the repeated samples of

size N, 95% of the generated confidence intervals will include the true value of Y. The

confidence interval was obtained through the formula:

95% CI for predicted Y’ = Y’ +/- tcrit * SEest

To measure the closeness of fit and obtain a numerical value that indicates the

proportion of variation of consumption that is predicted by the variation of disposable

personal income, U, R, %S&P, OPEC, and oil shocks, it is necessary to calculate an R2

value. The R2 was obtained by the formula:

= 1 ∑Y ′

∑ 2

Or R2 = Explained Variance/Total Variation


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The formula is the ratio of unexplained variation in the dependent variable

divided by the total variation of the dependent variable; subtracted from one. The value

of R squared tells us the proportion of variation in real consumption per capita

(dependent variable) that can be explained by real income per capita, %S&P, U, R,

Opec, Shock (independent variables). R2 has a range from 0 to 1. If the R2 value is 1; this

would indicate that 100% of the variation in consumption is explained by real income or

the other independent variables; a perfect fit. Likewise, if the R2 obtained is 0; this

would indicate that 0% of the variation in consumption is explained by real income and

the other independent variables. To make predictions from our model, the F-test and t-

test were performed. To better assess the value of R2 it is necessary to calculate the

adjusted R2.

The adjusted R2 is the R2 adjusted for the number of independent variables in the

equation. This adjustment gives the proportion of variability in Y that is predicted by

the regression equation, after taking into account the number of independent variables.

The adjusted R2 was obtained by the formula:

= 1 1 R1

1

Or R2 adjusted = proportion of variability in Y that is predicted by the regression

equation, after taking into account the number of independent variables.


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In regression analysis, the F-test is a one tailed test and assesses if the R2 is

statistically significant. The test is done by obtaining the F critical value and the F

calculated value. The F calculated value is obtained through the formula:

=∑Y′ / K 1∑ ′2/

This formula takes explained variation and divides this by the unexplained

variation. What this means is that the more variation in the observed data that is

explained by the regression line, the greater the F value is. The F critical value is a

particular value on the F-distribution table that is identified by a certain significance

level. The F-critical value can be obtained by looking at a F-distribution table at a certain

significance level (.05), finding the degrees of freedom in the numerator by the formula

K-1 where K represents the number of coefficients (including Y intercept) and finding

the degrees of freedom in the denominator N-K, where N represents the sample

number and K is the number of coefficients in the function. If the F calculated value is

greater than or equal to the F critical value, it is necessary to reject the Ho and affirm the

Ha; the function is statistically significant. If the F calculated value is less than the F-

critical value, the Ho is not rejected; the function is not statistically significant.

Unlike the F-test where we are testing the entire function, a t-test tests each

coefficient separately to see if each individual coefficient is significantly different from 0

at a certain significance level. Like the F-test, the t-test has two components a t-


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calculated value and a t-critical value. The T-calculated value is obtained through the

formula:

=

To get the t-calc, the coefficient is divided by the standard error of the coefficient.

The t-calculated value must be compared to the t-critical value. In this paper, the t-test

is a two tailed test; when looking for the T-crit value at a significance level of .05, it is

necessary to obtain the t-crit value at a level of .025 because the probability is split

between both tails. The T critical value can be found on a T sampling distribution by

obtaining the degrees of freedom through the formula N-K where N is the sample size

and K is the number of coefficients (including the intercept) at a two tailed significance

level of .025. If the │T-calc│is ≥ the │T-critical value│ the coefficient is statistically

significant, and it is necessary to reject the Ho and affirm the Ha. If the │T-calc│is < │T-

crit,│do not reject the Ho. At the .05 significance level, we cannot rule out the possibility

in the case of coefficient ‘b’ that there is no relationship between X and Y.

The Durbin Watson test measures whether successive residuals vary in value in a

random (independent) manner, or, instead, in a systematic manner. The Durbin Watson

test looks at d for time-series data to draw conclusions as to whether there are positive

serially correlated errors, or negative serially correlated errors. Positive serially

correlated errors indicate that a positive residual is followed by a positive residual or a

negative residual is followed by a negative residual. Negative auto correlation indicates

that a positive residual is followed by a negative residual or a negative residual is


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followed by a positive residual. This often happens as data values for one time period

are often correlated to values in the next successive time period. To test this, it is

necessary to obtain a d-critical value for a lower limit (dL) and another d-critical value

for the upper critical limit (du). If at the .05 level of significance d 4-dL it is

appropriate to reject the null hypothesis. If at a .05 level of significance d>du but d< 4-du

do not reject the null hypothesis. If we reject the null hypothesis; there is significant

evidence of serial correlation. Alternatively, if we do not reject the null hypothesis, there

is not significant evidence of serial correlation at the .05 level.

Table 3: The formulas

=Σ(Y Y′ )

95% CI for predicted Y’ = Y’ +/- tcrit * SEest

= 1 ∑Y

∑

The R2 formula assesses the ratio of unexplained variation in the dependent variable (c) divided

by the total variation of the dependent variable; subtracted by one.

= 1 1 R1

1

=∑Y′ / K 1

∑ ′

/

F-calc takes explained variation and divides this by the unexplained variation.

t-Calc = =

The coefficient is divided by the standard error of the coefficient.


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Data

The data for this regression analysis was acquired from the Federal Reserve in St.

Louis. The data is from 1962-1994, accounting for 33 observations. These data were

selected to ensure cyclical consistency. Both starting and end dates are neither trough or

peak years, but rather fall in expansionary periods in the economy (National Bureau of

Economic Analysis). The data for consumption and income is per capita consumption

and real per capita personal disposable income in chained 2009 dollars (US. Bureau of

Economic Research). To economists, the term chained means the data is adjusted for

inflation allowing for substitution between quantities of goods purchased. This is an

appropriate adjustment for inflation, instead of using the CPI index. Using chained

instead of CPI is more effective because it adjusts for changes in the goods a household

may buy in a given year. For example, as prices change, it’s not always fruit, vegetables,

and oil etc.; it may be fruit, vegetables, and computers. Both the data for Interest rates

and stock portfolio wealth are available in nominal terms, these are adjusted for

inflation using the consumer price index (CPI).

In this paper, Consumption Expenditures are real per capita consumption. Over

time, real per capita consumption expenditures have steadily increased every year with

consumption only decreasing in the year of 1974, 1980, 1991. To gain a richer

understanding of how the per capita consumption expenditure has changed over time;

it is more effective to look at how the consumption expenditure has changed each year

(see Table 4).


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Table 4: Per capita consumption expenditure changes

In Table 4, it is important to point out an intriguing characteristic. In the years

following the decrease in consumption, the following 2 to 3 years later, consumption

Years

Change

in Cons.

62-63 285

63-64 501

64-65 584

65-66 545

66-67 239

67-68 610

69-70 371

70-71 356

71-72 72672-73 604

73-74 -275

74-75 199

75-76 719

76-77 523

77-78 561

78-79 220

79-80 -260

80-81 84

81-82 80

82-83 83983-84 808

84-85 836

85-86 657

86-87 513

87-88 692

88-89 430

89-90 207

90-91 -248

91-92 524

92-93 491

93-94 616


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increased dramatically; perhaps to make up for the loss of consumption in that year. It

is also important to note that consumption is always less than disposable income.

The Disposable Income data is real per capita disposable income. Overtime, real

per capita disposable income has followed the same trend as consumption. From 1962

to 1994 disposable income has steadily increased only to drop during and shortly after

the OPEC oil embargo in 1973 to 1974 and during the oil shocks that occurred from 1979

to 1980 and 1990-1991.

The data used to measure Interest Rate was the annual bank loan prime rate. The

prime rate is the interest rate the people with the strongest credit receive. Overtime the

interest rate will start at a particular rate, undergo small fluctuations for about 2 to 3

years and then finally hit another number and continue this process. This pattern is not

true for the OPEC oil embargo and oil shocks. During these times; the interest rate

drastically increased from 5.25 in 1972 (the year before the embargo) to 8.03% in 1973

and 10.80% in 1974. During the oil shocks of in 1979 and 1980, the interest rate

significantly increased from 9.05% in 1978 to 12.66% in 1979 and 15.28% in 1980. The

interest rates response to the oil shocks in 1979 and 1980 was different from the shocks

experienced from the embargo. After the embargo was over in 1974, the interest rate

dropped approximately 3 percentage points in 1975, however after the oil shocks were

over in 1980, the following year the interest rate continued to increase from 15.28% to

18.86% in 1981, and finally decreased in 1982. There is an intriguing trend in the data. In


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the presence of an oil shocks, the interest rate response does behave the same, in a

slightly predictable way. This is, from the time the market reacts to the oil shock

Unemployment was measured as the Civilian Unemployment rate as a percent of

the labor force. Overtime, the unemployment rate has increased, but not very much. In

1962, the start of my data the unemployment rate was 5.50%. In 1994 the ending year of

my data, the unemployment rate was 6.10%; less than one percentage point increase.

The unemployment data did increase substantially after the OPEC embargo and the oil

shocks and take a greater time to decrease than consumption or income. Following the

years of the embargo and the oil shocks, unemployment rose to nearly 8.50% in 1975

and slowly decreased until the U.S. was hit with the oil shocks. In the years following

the oil shocks in 1979-1980 unemployment rose from 7.60% in 1981 to unemployment

levels of 9.70% in 1982 and 9.60% in 1983. Interestingly, when the oil shocks in 1990 and

1991 hit, unemployment rose slightly by .7% in 1992 and then in 1993 began to reduce

back to normal levels of unemployment. This was a much faster rate of reduction from

the shocks in 1979 and 1980 where the levels of unemployment increase for three years

until they started to reduce again.

Stock Portfolio Wealth data was acquired from professor Damodaran at NYU.

His data are consistently close to the % change in the S&P 500 index, using end-of-year

values.


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OPEC oil embargo in the years of 1973 and 1974 was measured by creating

dummy variables where a 1 indicates the embargo and a 0 indicates no embargo.

Approximately 6.6% of observations (N) received a 1.

Other Oil Shocks in 1979, 1980, 1990, 1991 were also measured by creating

dummy variables where 1 indicates an oil shock and a 0 indicates no oil shock.

Approximately 12.12% of the observations (N) received a 1.

Table 5 will present descriptive statistics of both the dependent variables and

each independent variable.

Table 5: Descriptive Statistics

Mean Std. Deviation NConsumption $17341.7273 $3928.21187 33

DI (USD) $19887.7273 $4252.04841 33

R 3.4982% 2.49844% 33

%S&P 5.8667% 15.71229% 33

eU 6.1636 1.55057% 33

Table 6: Descriptive Statistics for Dummy Variables

Variable Name Define % of Observations that are 1

And % of observations that are 0

gOPEC 1 = embargo

0 = no embargo

1 = 6.6%

0 = 93.4%


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hSHOCK 1 = shock

0 = no shock

1 = 12.12%

0 = 87.88%

Figure 1 shows the observed data point’s from1962-1994. The scatter plot above

shows a tight, evenly distributed data. Figure 1 suggests a positive linear relationship

indicating that with more income comes more consumption.

Figure 1: Consumption as a function of income, Scatterplot


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Results

The results for five regressions are presented in table 7.

Table 7: Estimated Regressions and Statistics1 2 3 4 5

a

-

995.31

3

-

689.54

2

2762.3

88

2123.5

62

2022.4

73

(-

4.715)*

(-

3.345)* 3.77 2.796 2.802

y 0.922 0.933 0.526 0.615 0.628

(88.768

)*

(85.097

)*

(6.932)

*

(7.305)

*

(7.873)

*

y2

9.99E-

06

7.89E-

06

7.55E-

06

(5.256)

*

(3.797)

*

(3.836)

*

R -0.168 -8.532

(-

0.009)*

(-

0.575)*

U -79.503 -34.933 -44.814

(-2.73)*(-1.323)*

(-1.894)*

%S&P -1.387 -2.022

(-

0.457)*

(-

0.818)*

OPEC

-

528.31

6

-

403.27

4

-

323.59

2

(-

2.701)*

(-

2.486)*

(-

2.488)*

Shock -5.305 25.008

(-

0.043)*

(0.251)

*

R2 0.996 0.998 0.998 0.998 0.998


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Adj R2 0.996 0.997 0.998 0.998 0.998

F

7879.7

50

1799.0

84

7338.2

66

2339.8

12

4401.0

63

d 0.671 1.075 1.284 1.616 1.545

N 33 33 33 33 33

t= ()*

Regression 1 The simple linear Keynes consumption model: F-critical value is obtained

from the F distribution table. In this model there is 1 degree of freedom (K-1) in the

numerator and 31 degrees of freedom (N-K) in the denominator this generates a 4.16 F-

critical value (wmich.edu). The F calculated value is 7,879.750. From this comparison we

can reject the null hypothesis and accept the alternative hypothesis because the F

calculated value is greater than the F-critical value. This means that with a 95% level of

confidence, we conclude our consumption function is significantly different from zero.

The t-critical value was obtained from the T distribution table. From the table of

.05 there are 31 degrees of freedom; the T-critical value at this significance level is 2.040

(math.wika.com). For coefficient ‘a’ the t-calculated value is 4.715 and for the ‘b’

coefficient the t-calculated value is 88.768. After performing the t-test, both |t values|

are greater than the |t-crit|, indicating that coefficients ‘a’ and ‘b’ are significantly

different from zero; The Ho is rejected and the Ha is affirmed.

The coefficient ‘b’, representing income explains the change in consumption in

relation to the change in disposable income. In more tangible terms, because the sign is

positive and is statistically significant, this means that for the years of 1962 to 1994, for


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every additional dollar of real disposable income earned; on average people consume

$.922 of that dollar.

For coefficient ‘a’ (autonomous consumption); the expected sign was positive,but it is in fact negative. This is an intriguing discovery. This means that if real per

capita income is zero; people spend $995.313, which has no economic meaning.

The R2 for regression 1 was .96. This means that 96% of the variation in

consumption is explain by real per capita disposable income. The adjusted R2 is also .96,

which means after accounting for the amount of independent variables in the model,

96% of the variation in consumption is explained by real per capita disposable income.

The Durbin Watson hypothesis test (which tests for serial correlation) was

conducted. The dL critical value for regression 1 is 1.383 and the du value is 1.1.508. The

d calculated value is .671, there for because d < dL at the .05 level; there is significant

evidence of serial correlation. The conclusion for regression one is that a re-estimation

of consumption function is warranted to address problems that might be due to omitted

variables or functional form.

Regression 2 added the variables R, U, %S&P, OPEC, and SHOCK to help better

explain the changes in consumption. In this regression, the adjusted R square was .997 a

significant increase from .96 in regression 1. The F-critical value for regressioin 2 was

2.4741 which is greater than the F-calculated value 2067.536. It is appropriate to reject

the null hypothesis and affirm the alternative, indicating that the regression

significantly different from zero. After accounting for problems related to omitted


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variables, there is a significant improvement in the durbin-watson d statistic. The

Durbin-Watson d statistic increased from .671 to 1.075 which is an improvement closer

to 2. Because the d is not < dL (1.061) or d is not > 4- dL (2.939) it is not appropriate to

reject the null hypothesis and because d is not > du (1.508) at the .05 level, it is also not

appropriate to not reject the null hypothesis. In result of our inability to neither reject

the null hypothesis or not reject the null hypothesis at the .05 level; the test is

inconclusive. In regression 2 our expected sign for the constant was positive, however

it is negative. Regression 2 yielded expected positive signs for income, expected

negative sign for interest rate, expected negative sign for unemployment, expected

positive sign for wealth, expected negative sign for OPEC embargo; but yielded an

unexpected negative sign for oil shocks.

The T-test was conducted to test the statistical significance of the coefficients.

The t-critical value as obtained from a t-distribution table (stat.tamu.edu). At the .05

level of significance, the t-critical value was obtained with 26 degrees of freedom;

yielding a t-critical value of 2.056. At the .05 coeffecients a, Y, U, and gOPEC are all

significantly different from zero. However, coefficients: R, %S&P, hSHOCK are not

significantly different from zero. It is notable to indicate that coefficient %S&P t-

calculated value was 2.027 very close to the t-critical value of 2.056.

To identify the possibility of muticollinearity (independent variables are highly

correlated with other independent variables) please see Pearson correlations matrix

(appendix). It is determined that DI has multicollinearilty to consumption, R and eU. R


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has multicollinearlity with consumption, DI, %S&P, eU, and gOPEC. Coeffecient

‘%S&P’ has multicollinearilty with R, eU, gOPEC, and hSHOCK. Coeffecient ‘eU’ has

multicollinearilty with consumption, DI, R, and %S&P. Coeffecient ‘gOPEC’ has

milticollinearilty with R and %S&P. Lastly, coefficient ‘hSHOCK’ has evidence of

multicollinearilty with consumption, DI, and %S&P.

In conclusion, for regression 2 adding in omitted variables has improved the

regressions ability in explaining changes in consumption; however it is appropriate to

check the functional form of the data.

Regression 3 was conducted to see if the regression improved as we changed the

functional form. Regression 3 is a regression of consumption as a function of a quadratic

specification of Y to reflect evidence of a curved relationship between consumption and

Income. The adjusted R2 for regression 3 was .998 a significant improvement from .96 in

regression 1.

At a .05 level of significance with 2 degrees of freedom in the numerator and 30

degrees of freedom in the denominator the F-critical value is 3.3158. The F-calculated

value is 7338.266; thus the model is significantly different from zero. There has been an

improvement in the d statistic from regression 1. The d statistic for regression three is

1.284. The critical dL is 1.321

To confirm the curvature relationship, it is necessary to conduct a t-test on

coefficient ‘c’ (coefficient for Y2). At a significance level of .05 with 30 degrees of

freedom in the numerator, the t-critical value is 2.042; because the t-calculated value for


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coefficient ‘c’ is 5.256, we can conclude that ‘c’ is statistically significant and confirm the

curvature in the relationship between consumption and income. The signs for

coeffecients ‘b’ and ‘c’ (the coefficients for Y and Y2) are both positive, indicating that Y

has increased from 1962-1994 and C has increased at an increasing rate.

Regression 4 adds R, U, %S&P, gOPEC, hSHOCK to the quadratic specification

of Y. The adjusted R2 for regression 4 is .998; no improvement from regression 3 and

only one one-hundredth of a percentage point increase from regression 2. The F-critical

value is 2.4047 and the F-calculated value 2282.138; indicating that the model is

statistically significant. There was an improvement in the Durbin-Watson test statistic,

which is 1.580. However, the Durbin-Watson hypothesis test is inconclusive at the .05

level of significance. To confirm the curvature, a t-test is conducted. At the .05 level with

29 degrees of freedom the t-critical value is 2.045. The t-calculated value for coefficient

‘c’ (coefficient for Y2) is 2.909; with a .05 level of significance we can conclude that

coefficient ‘c’ is statistically significant and the curvature is confirmed between

consumption and Y. The signs for coefficients ‘b’ and ‘c’ (coefficients for Y and Y2) are

positive indicating that Y has increased from 1962-1994 and consumption has increased

at an increasing rate. In this regression, coefficients R, %S&P, eU, hSHOCK are all

statistically insignificant. Coefficient gOPEC is statistically significant.

Regression 5 drops variables R, Shock and %S&P; in an effort to improve statistical

significance of the economic variable unemployment. The adjusted R2 is .998, which is

the same as regression 4. The t-critical value is 2.048, indicating that disposable income


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and OPEC are statistically significant at the .05 level of significance (t-calc 7.873 and

2.488), but is statistically insignificant (1.894). Although unemployment is statistically

insignificant, there was an improvement in the t-calculated value, after we dropped

%S&P. The t-calc increased from -1.637 to -1.894; indicating that there is some

relationship between consumption and unemployment; as the Keynesian consumption

model suggests. The F-calc for regression 5 is 4401.063 and the f critical value is 2.71,

indicating that the regression is statistically significant. The d statistic for regression 5 is

1.545, which is not an improvement from regression 4. For regression 5, d is not less

than dL (1.196), not greater than 4-dL (2.807) and not greater than du (1.730); we can

conclude at a .05 level of significance that the Durbin Watson test is inconclusive.

Although the d for regression 5 did not show an improvement from regression 4, it is

negligible, as both tests are inconclusive. The signs for coefficients ‘b’ and ‘c’

(coefficients for Y and Y2) are positive indicating that Y has increased from 1962-1994

and consumption has increased at an increasing rate. No independent variables have

correlations of .7 or higher, indicating there is no evidence of severe multicollinearity

between independent variables.

Interpreting Regression Coefficients

The best regression that will allow for the best predictive analysis is regression 5.

Constant for regression 5 is $2022.473. This means, in the presence of the variables:

disposable Income, unemployment and OPEC, real per capita consumption in the

United Sates is $2022.473.


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Unemployment coefficient for regression 5 is $-44.814. This means for every 1

percentage point increase in the economic variable unemployment, real per capita

consumption in the U.S. economy for the years 1962-1994 decreases by $44.814.

OPEC coefficient for regression 5 is $-323.592. This means for every OPEC oil shock,

consumption decreases by 323.592.

Disposable Income: MPC

Regression 1 (C = a + bY) coefficient b for real per capita personal disposable income is

$.922. This means for every additional dollar off real per capita personal disposable

income, people spend $.92 of that dollar. Alternatively, people save about 8 cents of

every additional dollar.

Regression 2: C = a + bY + dR + eU + f%S&P + gOPEC + hSHOCK coefficient b for real

per capita personal disposable income is $.933. This means for every additional dollar of

real per capita personal disposable income, people spend about 93 cents.

Regression 5: C = a + by + eU + gOPEC coefficient b (MPC) varies at each point along

the function. Taking the first derivative of C with respect to Y the MPC for the year 1962

is .815, for the year 1982 the MPC is .938, and for 1994 the MPC is 1.028. This means in

1962 for every additional dollar personal disposable income, people will spend roughly

82 cents. In 1982, for every additional dollar of personal disposable income, people will

spend roughly 94 cents. Lastly, in 1994, for every additional dollar of personal

disposable income, people will spend roughly 1.3 dollars. The general trend of the MPC

is a gradual increase over time, increasing 12 cents from 1962-1982 and 9 cents from


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1982-1994. The MPC for regressions 1 and 2 were very close only increasing by 1 cent

and the year 1982 was also close differing only 1 cent from regression 2 and two cents

from regression 1. However, there is a high degree of variance in the MPC for

regression 1 and 2 and the years calculated for regression 5 (1962, 1982,1994). The MPC

decreased by 11 cents in 1962 from regression 1 and 12 cents from the MPC in

regression 2.

Simple Multiplier

The multiplier predicts the change in Income for a given change in government

spending (G), planned Investment by businesses (Ip) or aggregate production (Ap). The

multiplier was calculated by the following formula:

Multiplier =

−

The multiplier for regression 1 is 12.5 and the multiplier for regression 2 is 14.92. For

regression 5, the multiplier for the year 1962 is 5.42 and the multiplier for 1982 is 16.37.

The MPC for 1994 is 1.3, which in theory is possible, however the resulting multiplier

calculated with this MPC would be a negative number; which has no economic

meaning. Resulting from the nature of the multiplier equation; as the MPC increases, so

does the multiplier. For the year 1962 any change in G, Ip or Ap will result in an

increase in income by 5.42 times the amount of the change and for 1982 it will increase

income by 16.37 times.


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Table 8. Estimated Marginal Propensity to Consume (MPC) and Simple Multiplier

Regression 1: C = a + bY

Regression 2: C = a + bY +dR + eu

f%S&P + gOPEC

MPC = .922

MPC =

.933

Multiplier = 12.5

Multiplier

= 14.92

Regression 5: C = a + bY + Y2 + eU + OPEC

MPC = b +2cY

Year MPC Multiplier

1962 0.815 5.42

1982 0.938 16.37

1994 1.028 N/A

Extrapolation

Extrapolation uses the model presented in table 9 to forecast predictions about

future consumption expenditures.

In 1995, the observed Y was $27,180 and Unemployment was 5.6%; this was not an

OPEC embargo or oil shock year. The regression equation predicts for these values that

consumption would be:

C’ +/- tcrit* SE est = 24414.441 +/- 2.048 * (167.34625)


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Thus, the regression predicts with 95% confidence that consumption would likely fall in

the range of $24,071.68 to $24,757.20.

In 1996, the observed Y was 27719 and unemployment was 5.4%; this was not an

OPEC embargo or oil shock. The regression equation predicts for these values that

consumption would be

C’ +/- tcrit* SE est = 25,047 +/- 2.048 * (167.34625)

Thus, the regression predicts with 95% confidence that consumption would likely fall in

the range of $24,642.40 to $25,327.92.

The observed value for consumption in 1995 was $24,485 and the observed value

for consumption in 1996 was $25,047; both of which fell within the respective 95%

confidence interval.

As shown in table 9, the 1995 prediction underestimates the consumption by

.29% and the 1996 prediction underestimates the consumption by 1.62%. Based on how

close the prediction was to the observed level of consumption, the regression

predictions are very precise.


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Table 9. forecasting levels of per capita consumption for the years 1995 and 1996 using RealPersonal Disposable Personal Income, Unemployment Rate, and OPEC Oil Embargo

Regression 5: C = a + bY + Y2 + eU + gOPEC

Year 95% Confidence Interval Observed

1995 $24,071.68 to $24,757.20 $24,485

1996 $24,642.40 to $25,327.92 $25,047----------------------------------------------------------------------------------------------------------------------------------

year Predicted Observed Percent difference

1995 $24,414.44 $24,485 -0.29%

1996 $24,642.40 $25,047 -1.62%

Conclusion

The model (C = a + bY + Y2 + eU + gOPEC) is an accurate model to draw

conclusions and make predictions about future consumption levels. In the year 1995 the

predicted level of consumption was $24,414.44 and the observed level of consumption

was $24,485. The regression underestimated the predicted level of consumption by

.29%. In the year 1996 predicted level of consumption was $24,642.40 and observed level

of consumption was 25,047; the regression underestimated the predicted level of

consumption by 1.62%


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http://www.nber.org/

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[A229RX0A048NBEA], retrieved from FRED, Federal Reserve Bank of St. Louis

https://research.stlouisfed.org/fred2/series/A229RX0A048NBEA/, February

11, 2015.

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Pledged: Erik Robinson

http://homepages.wmich.edu/~hillenbr/619/AnovaTable.pdfhttp://math.wikia.com/wiki/T_tablehttp://math.wikia.com/wiki/T_tablehttp://math.wikia.com/wiki/T_tablehttp://www.nber.org/http://www.nber.org/http://www.nber.org/http://math.wikia.com/wiki/T_tablehttp://homepages.wmich.edu/~hillenbr/619/AnovaTable.pdf


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Documents

Paper 2 Regression Analysis