Download The effects of Bank Prime Loans Rate and Unemployment Rate on

The effects of Bank Prime Loans Rate and Unemployment Rate
on the US Public Debt during from 1949 to 1980
Alexandre Catsicas, CBA1, Fordham University, Bronx, New York 10458,
[email protected]
Abstract
The public debt is an issue that every nation has to face. It can be considered as the total
nation’s debts; an indicator of how much public spending is financed by borrowing instead of
taxation, and which at the end, can be indirectly interpreted as a debt of the citizens. In this work,
we will explore the interesting effects of two different socio-economic phenomenon such as the
Bank Prime Loan Rate (X) and the Unemployment Rate (Z), and the, try to see if their
consequences can be related to the Public Debt of the United States (Y) between 1949 and 1980.
I. Introduction
When watching the news or reading the newspaper, it has been difficult not to be exposed
to a topic discussing about the economical crisis we are facing at this very moment. Terms like
“prime loan rate”, “unemployment rate”, as well as “public debt” of a nation, can become issues
that will most likely affect each of us in a more or less direct way. In this paper, we will
concentrate during the years between 1949 and 1980, in order to try to build a model where the
potential correlations between these three terms will be explored, and which at the end will lead
us to the elaboration of the kind of connections that can be established within our three variables.
Regression, as a way that will help us to break down the data into a form that can provide the
desired connections and correlations between the data, will be particularly helpful in our
research. Indeed, it is the regression that will allow us to prove if prime loan and unemployment
rate can have particular consequences on the public debt of a nation. In this work, we will
explore some of the mechanisms of the Economy and try to see if it can be statistically be proved
that the higher the unemployment and bank prime loan rates have been in the US from 1949 to
1980, the higher the public debt of the country has been.
1
This term paper will be written in March/April 2009 in partial fulfillment of Statistical Decision Making course by
Prof. Vinod
II. Model and data
As discussed earlier, the model proposed in this work will evaluate the eventual possibility
of a causal relationship between a nation’s unemployment rate (Z) and bank prime loan (X) on
that nation’s public debt (Y). For this, we shall proceed with a linear regression model which will
be represented by :
Y=β0+β1X+ β2Z+
This formula will represent the true population regression equation, in which  stands for
the unknown errors in our model. Once plugged into R, the estimated model becomes :
Y=b0+b1X+b2Z+residual
Where Y, the public debt of a nation, would be our dependent variable, caused by our
independent variables X and Z, the bank prime loan and the unemployment rate respectively.
The whole set of data that we will analyze in this work has been taken from the following
website : www.economagic.com .
The United-Sates’ public dept (Y) is the amount of money owed by the Federal
Government of the US to holders of US debt instruments. In other words, the public debt is the
amount of money owed by the Government to its creditors, whether they are nationals or
foreigners. The bank prime loan (X) is a term which designates a reference interest rate used by
banks. This interest rate at which banks will lend to their customers is a very important reference,
for it can be used as an index in calculating rate changes to adjustable rate mortgages and other
variable rate implicating short term loans, such as private student loans for example. The
unemployment rate (Z) is commonly defined as the percentage of people in the labor force but
who are unemployed.
While looking at our variables, we can see that two of them have a distribution skewed to
the right. Indeed, Y and X have both a Median that is smaller than their respective Mean.
Whereas Z is skewed to the left with a Median slightly bigger than its Mean. Variable Y as a
mean of $39.32 (in tens of billion), with a variance of 327.47. Variable X has a mean of 5.77%
with a variance of 9.15, and variable Z has a mean of 5.32% with a variance of only 2.07 (all
values have been rounded up to two decimals for this description).
a) Units of measurement
My two independent variables, X and Z, are measured in the same magnitudes. Indeed,
both the unemployment rate and the bank prime loan are expressed in percentage. For practical
reasons, such as an easier reading of the analysis and an easier manipulation of the data, the
liberty to round up each initial value to one decimal has been taken in this work. Regarding our
dependent variable Y, the public debt, the values were initially expressed in hundreds of billion.
Adjustments were made, and each value of Y has been divided by 1’000’000’000. The values
have then been rounded up once again to one decimal in order to facilitate the lecture and the
manipulation. Since not all the variables will be of the same magnitude, we are to expect that the
visual plots will be slightly skewed, but at the end, the regression analysis should remain the
same.
b) Basic descriptive statistics
In this work, all the data analysis has been performed at the 95% confidence interval. The
following description contains value that have all been rounded up to two decimals.
Variable Y : the public debt, has a mean of 39.32, the standard error is 3.2, the median is
31.45, the standard deviation is 18.1, the sample variance is 327.47, the kurtosis is 1.23, and the
skewness is 1.52.
Variable X : the bank prime loan, has a mean of 5.77, the standard error is 0.53, the
median is 4.65, the standard deviation is 3.02, the sample variance is 9.15, the kurtosis is 1.54,
and the skewness is 1.32
Variable Z : the unemployment rate, has a mean of 5.32, the standard error is 0.25, the
median is 5.5, the standard deviation is 1.44, the sample variance is 2.07, the kurtosis is -0.75,
and the skewness is 0.21.
From the data analysis and descriptive statistics, we can see that the multiple R=0.8978,
R =0.806, and adjusted R2 =0.7926 are all under one, as they should be (by definition), but also
pretty high. The adjusted R2 suggests that our model is about 79.26% successful in explaining
the United-States public debt. See Appendix of this paper.
2
y
x
z
nobs
32.000000 32.000000 32.000000
NAs
0.000000
0.000000
0.000000
Minimum
25.300000
2.000000
2.900000
Maximum
90.800000 15.300000
8.500000
1. Quartile
27.350000
3.800000
4.250000
3. Quartile
43.475000
7.075000
6.025000
Mean
39.318750
5.765625
5.315625
Median
31.450000
4.650000
5.500000
Sum
1258.200000 184.500000 170.100000
SE Mean
3.198977
0.534714
0.254401
LCL Mean
32.794393
4.675068
4.796770
UCL Mean
45.843107
6.856182
5.834480
Variance
327.470605
9.149425
2.071038
Stdev
18.096149
3.024802
1.439110
Skewness
1.521094
1.319349
0.212147
Kurtosis
1.126328
1.541404 -0.754337
We can also test the Correlation Coefficients :
For X and Y, the Correlation is 0.8655, with a p-value of 1.620e-10. The p-value being so
small, we can reject the null hypothesis and therefore conclude that there is a good correlation
between the public debt and the bank prime loan.
For Z and Y, the Correlation is 0.4998, with a p-value of 0.003586. Since the p-value is
quiet small, we can also reject the null hypothesis, but this time the correlation between the
public debt and the unemployment rate will not be as strong as between the public debt and the
prime loan.
For X and Z, the Correlation is only 0.316, with a p-value of 0.07805. Since the p-value
remains small, we can also reject the null hypothesis. But we can see that the correlation between
the unemployment rate and the bank prime loan is the smallest of our three relationships.
y
x
z
y 1.0000000 0.8655309 0.4997698
x 0.8655309 1.0000000 0.3160369
z 0.4997698 0.3160369 1.0000000
c) Outlier detection for X, Y, and Z
IQR=Q3-Q1
LOW=Q1-1.5*IQR
UPPER=Q3+1.5*IQR
The outlier detection limits for Y are :
Q1-1.5*(inter quartile range)= 3.15500000000001
Number of outliers below it are= 0
Q3+1.5*(inter quartile range)= 67.675
Number of outliers above it are= 4
Actual values above the upper limit are: 70, 77.2, 82.7, 90.8
The outlier detection limits for X are :
Q1-1.5*(inter quartile range)= -1.1125
Number of outliers below it are= 0
Q3+1.5*(inter quartile range)= 11.9875
Number of outliers above it are= 2
Actual values above the upper limit are: 12.7 and 15.3
The outlier detection limits for Z are :
Q1-1.5*(inter quartile range)= 1.5875
Number of outliers below it are= 0
Q3+1.5*(inter quartile range)= 8.6875
Number of outliers above it are= 0
Even though we do have four outliers in the variable Y, and two outliers in the variable
X, the fact that our adjusted R2 is so high means that the outliers should not provoke any
misreading in our future results.
Scatter Plots:
The relationships between our two independent variables (X and Z) and the dependent
variable (Y) will be shown in the following scatter plots.
Figure1 shows us the relationship between the public debt and the bank prime loan. The
positive correaltion appears pretty strong, with a regression line of almost 45 degres. Since our
two varaibles are increasing at an almost similar rate, we can conclude that there is a significant
relationship between Y and X.
80
70
60
50
40
30
Public Debt (in tens of billion)
90
Fig. 1: Public Dept versus Bank Prime Loan Rate
2
4
6
8
10
Bank Prime Loan Rate
12
14
Figure2 shows us the relationship between the public debt and the unemployment rate.
The scatter plott on itself is quiet spread and doesn’t seem to indicate a lof of information. But
once the regression line is added, we can see positive relationship between Y and Z. But this
positive correlation still remains smaller than the one for Y and X.
80
70
60
50
40
30
Public Debt (in tens of billion)
90
Fig. 2: Public Dept versus Unemployment Rate
3
4
5
6
Unemployment Rate
7
8
Figure 3 will depict the relationship between the residuals against the bank prime loan
rate, and Figure 4 will depict the relationship between the residuals against the unemployment
rate. Ideally, there should be no relations in the plots for the model to be significant, which is our
case.
0
-10
Residuals
10
20
Fig. 3: Residuals against Bank Prime Loan Rate
2
4
6
8
10
12
14
Bank Prime Loan Rate
0
-10
Residuals
10
20
Fig. 4: Residuals against Unemployment Rate
3
4
5
6
Unemployment Rate
7
8
Our last plot, Figure 5, will depict the actual Y (the public debt) compared to the fitted Y.
Although we do not have a perfect linear correlation, we can see that the different values still
seem to follow a certain constant pattern. We have to remember that our model is trying to
explain a very complicated macroeconomic mechanism, a mechanism which as the Figure 5
tends to show us, will involve much more independent variables than the only two that we are
working with. Nevertheless, let us remark that an almost perfect virtual 45 degree line can still be
perceived in this model.
60
40
20
Fitted Y
80
Fig. 5: Fitted against Actual Y
30
40
50
60
Actual Y
70
80
90
Our Fitted Regression equation is :
Y=b0+b1X+b2Z
Where b0=-4.5960, b1= 4.7029, and b2= 3.04
Once these figures are plugged into the equation we get :
Y=- - 4.596 + 4.7029X + 3.1604Z + residual
Intercept
X
Z
Coefficients
Standard Error
-4.5960
5.7739
4.7029
0.5157
3.1604
1.0840
t Stat
-0.796
9.119
2.915
Pr(>|t|)
0.43250
5.13e-10
0.00678
The “Coefficients” column gives us the regression intercept and slope coefficients b1and
b2. The slope b1 measures the change in Y from a unit change in X. The slope b2 measures the
change in Y from a unit change in Z.
Regression Statistics
Multiple R Squared
Adjusted R Squared
Standard Error
Observations
0.806
0.7926
8.241
32
As described earlier, the “Regression Statistics” are in rather high, confirming the
reasoning behind our model. Both of our Multiple R are under 1.0, as they have to be. The
standard error is 8.241. The degree of freedom is 29 and the regression degree of freedom is 2.
Now in order to determine how close a line is to the data, it becomes necessary to
measure the predicted value of Y given X, versus the actual value of Y. The difference between
the observed actual Y value, and the predicted Y value is what call the model error. Therefore,
the estimated model will change to :
Ŷ =b0+b1X+b2Z+e
With the following fitted value :
Ŷ= - 4.596 + 4.7029X + 3.1604Z + 8.241
III. Statistical testing of the model
In order the validity of our model, we will conduct two T-tests and one F-test. For the Ttests, each of the regressor variables (X and Z) will be considered in relation to the dependent
variable (Y). The F-test will determine if both X and Z against Y can measure the overall fit of
data in describing the relationship between the bank prime loan rate and the unemployment rate.
Y Intercept
Prime Loan
Unemployment
Coefficients
-4.5960
4.7029
3.1604
Standard Error
5.7739
0.5157
1.0840
t Stat
-0.796
9.119
2.915
P-value
0.43250
5.13e-10
0.00678
Lower 95%
-16.405
-3.648
0.9433
Upper 95%
7.213
5.758
5.377
The first t-Test :
Null hypothesis H0: β1 = 0
Alternative hypothesis: β1≠ 0
Rule : Reject the Null if | t | > 2.045
The observed value of the t-statistic for β1 is 9.119 at a 29 DF. The critical value defines
the rejection region. If the observed value is greater than the critical value, then the null
hypothesis is rejected. Since the critical value at a 95% confidence level is 2.045, we can
therefore reject the null. We can conclude that the bank prime loan (X) has definitely had a
statistically significance on the public debt of the United-States between 1949 and 1980.
The second t-Test :
Null hypothesis H0: β1 = 0
Alternative hypothesis: β1≠ 0
Rule : Reject the Null if | t | > 2.045
The observed value of the t-statistic for β2 is 2.915 at a 29 DF. The critical value defines
the rejection region. If the observed value is greater than the critical value, then the null
hypothesis will again be rejected. Since the critical value at a 95% confidence level is 2.045, we
can therefore reject the null. We can conclude that the unemployment rate (Z) has also had a
statistically significance, but less notable this time, on the public debt of the United-States
between 1949 and 1980.
The F-Test :
The F-Test indicates whether or not the overall model is useful as a tool for explaining
the theory of our model. As seen previously, the null hypothesis states that the model is not
useful, whereas on the other hand, the alternative states that both of our independent variables
are useful.
Null hypothesis: H0: β1 = β2 = … βk = 0
Alternative hypothesis: Ha: at least some βi ≠ 0
For our model, the F-value is 60.24 and the p-value is 4.71e-11. Since we are testing our
model at the 95% confidence interval in which  = 0.05, we can reject the null because  > p.
Also, we could just have compared the F statistic to the critical value 3.328 in order to test our
hypothesis. Since the observed value of F is larger than the critical value, we can therefore reject
the null. We can conclude that both of our two independent variables are significant with positive
slope. Our model is therefore rather significant in explaining the effects of the bank prime loan
and the unemployment rate on the public debt of the United-States between 1949 and 1980.
Conclusion and final remarks :
The conduction of this work has been very interesting on several different points. The
Economy in which we are living in appears to be a very complex notion, to which it seems
difficult to attribute a defined spectrum of causes and effects. This work has allowed to establish
quiet strong links between three different factors. The statistical evidences that we have
discovered throughout this entire analysis, showed us the repercussions of a social factor such as
the unemployment rate, as well as the effects of a financial factor such as the bank prime loan
rate on the public dept of a nation. In both cases, we saw that an increase of our independent
variables X and Z caused an increase of the public dept of the United-States from 1949 to 1980.
Of course our model doesn’t allow us to clearly explain how these two independent variables act
on our dependent variable, but at least the model showed us that positive correlation exists.
If look back at our R2 (0.7926), our model is almost 80% effective in explaining the
consequences on the public debt. And here is a way where we can try to explain the
consequences of (X) and (Z) in the macroeconomic sphere :
According to our research, we can observe that the bank prime loan rate (X), had a t-Test that
could definitely allow us to reject the hypothesis stipulating that it had no effect on the public
debt. Which after all makes a certain sense when we remember one of the basic principles of
macroeconomics where the equilibrium state is defined by : Y = C + I + G + (EX-IM). Now we
can see that a decrease in C (Consumption Expenditure / Spending of Households) and a
decrease in I (Domestic Investment / Spending by Firms) will cause a decrease on Y. If we go
back to our variable (X), an increase of the bank prime loan rate could have the following effects
: if the interest rates at which households and firms usually borrow money increases, the general
consumption C, and the general investment I in the economy will decrease. But since our
equilibrium must be maintained, one way will be to increase G (Government expenditures). But
if G increases, it becomes inevitable that the public debt of the country will also increase due to
new expenses. The public dept is therefore clearly affected by our variable (X), the bank prime
loan rate. We could therefore imagine that if a country has a public debt becoming too important,
one solution would be to decrease the prime rates in order to give rise to a general increase of
households’ consumption and firms’ investment. Like this, our equilibrium will be maintained
without any increase of G.
Now regarding our second independent variable (Z), the unemployment rate, we have seen that
correlation was not as explicit as with the variable (X). Indeed, even if we have been able to
reject the null hypothesis with the help of the t-Test, the effects of the unemployment rate are a
little bit less noticeable than the bank prime loan rate. But at the end we can still establish a
positive correlation between this unemployment rate and our public debt. If you look again at
our equilibrium formula, we can deduct that an increase in the unemployment rate, will cause
households to consume less (with no job there is no money to spend), if the consumption C
decreases, one of the alternative to maintain the equilibrium could be taken by the government G
to spend more. Which once again, will cause an increase in the public debt.
At the end, our model can let us imagine that a good way to decrease the public debt of a
country by still maintaining the equilibrium state, would be a combination of a decrease in bank
prime loan rates, and a general increase in hiring the unemployed force of the population.
Appendix
Y=c(25.3, 25.7, 25.5, 25.9, 26.6, 27.1, 27.4, 27.2, 27.1, 27.6, 28.5, 28.6, 28.9, 29.8, 30.6,
31.2, 31.7, 31.9, 32.6, 34.7, 35.4, 37.1, 39.8, 42.7, 45.8, 47.5, 53.3, 62.0, 70.0, 77.2, 82.7, 90.8)
X=c(2.0, 2.1, 2.6, 3.0, 3.2, 3.1, 3.2, 3.8, 4.2, 3.8, 4.5, 4.8, 4.5, 4.5, 4.5, 4.5, 4.5, 5.6, 5.6, 6.3, 8.0, 7.9, 5.7, 5.2, 8 .0,
10.8, 7.9, 6.8, 6.8, 9.1)
Z=c(7.6, 5.2, 3.3, 3.0, 2.9, 5.6, 4.4, 4.1, 4.3, 6.8, 5.5, 5.5, 6.7, 5.6, 5.6, 5.2, 4.5, 3.8, 3.8,
3.6, 3.5, 5.0, 6.0, 5.6, 4.9, 5.6, 8.5, 7.7, 7.1, 6.1, 5.9, 7.2)
x
y
z
Min. : 2
Min. :25.3
Min. : 2.9
1st Qu.:3.8
1st Qu.:27.35
1st Qu.: 4.25
Median :4.65
Median :31.45
Median : 5.5
Mean :5.77
Mean :39.32
Mean : 5.32
3rd Qu.:7.08
3rd Qu.:43.48
3rd Qu.: 6.03
Max. :15.3
Max. :90.8
Max. :8.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.5960 5.7739 -0.796 0.4325
x
4.7029 0.5157 9.119 5.13 e-10 ***
z
3.1604 1.0840 2.915 0.00678 **
--Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' '
1
Residual standard error: 8.241 on 29 degrees of
freedom
Multiple R-Squared: 0.806, Adjusted R-squared:
0.7926
F-statistic: 60.24 on 2 and 29 DF, p-value: 4.71e-11
Residuals:
Min
1Q Median
3Q
Max
-16.127 -4.917 -1.259 3.426 20.996
Response: y
Df Sum Sq Mean Sq F value Pr(>F)
x
1 7605.0 7605.0 109.2567 3.59e-11 ***
z
1 577.2 577.2 8.2925 0.007548 **
I(x^2) 1 20.4 20.4
0.2929 0.592629
Residuals 28
1949
69.9
--Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' '
1
> confint(reg1)
2.5 %
(Intercept) -16.4049585
x
3.6480938
z
0.9433127
97.5 %
7.212971
5.757740
5.377483
data: x and y
t = 9.4652, df = 230, p-value = 1.620e-10
alternative hypothesis: true correlation is not equal to
0
95 percent confidence interval: 0.7402478
0.9327238
sample estimates:
cor 0.865531
data: z and y
t = 3.1603, df = 30, p-value = 0.003586
alternative hypothesis: true correlation is not equal to
0
95 percent confidence interval: 0.1829596 0.7225477
sample estimates:
cor 3160369
data: x and z
t = 1.8245, df = 30, p-value = 0.07805
alternative hypothesis: true correlation is not equal to
0
95 percent confidence interval: -0.03670165
0.59874863
sample estimates:
cor 0.3160369
CHECKLIST
PART 1 your name and title of the paper.
1) Is the title sufficiently descriptive? Is it typed centered, bold, properly capitalized, in large font and along the top
line?
0.5 points. I claim full credit for this 
2) Is there your Name, affiliation and E-mail along second and third line?
0.5 points. I claim full credit for this 
3) Is there the (*) Footnote with date, the phrase “partial fulfillment,” etc.
0.5 points. I claim full credit for this 
4) Is the Abstract LESS THAN 125 words and descriptive of the model? (Please do give data sources in the initial
version due march 9 but delete it from the final version of the abstract as attached to the paper due March 23, where
the data information goes in the data appendix)
2.5 points. I claim full credit for this 
5) Are there exactly four numbered sections with bold titles present? How many pages? (Cut a point if raw data are
NOT attached in the program and data appendix).
0.5 points. I claim full credit for this 
Points earned in PART 1 so far are:
4.5 Out of 4.5 .
PART 2
6) Are data sources mentioned? 0.5 points I claim full credit for this 
7) Are descriptive stats including correlation coefficients mentioned?
0.5 points I claim full credit for this 
8) Is outlier detection done for Y, X and Z variables?
1 point I claim full credit for this 
9) Are 4 figures present? Fig 1 has Y vertical &X horizontal (Y vs. X), Fig 2 has Y vs. Z,
Fig 3 has residuals vs X and Fig.4 has residuals vs. Z. Note that if the residuals do not show any particular pattern,
the model as a whole is good.
0.5 points I claim full credit for this 
10) Are the figures labeled correctly? 0.5 points I claim full credit for this 
11) Are the Correct variables represented on the vertical axis ?
0.5 points I claim full credit for this 
12) Are all four figures explicitly discussed in the text? 0.5 points I claim full credit for this 
Points earned in PART 2 so far are:
3.5 Out of 4
PART 3
13) Is regression equation with equation numbers reported? e.g.:
Y = 0.0123 + 2.3345 X  0.3567 Z + residual (numbers will be different for each)
14) Is it centered? It should be centered.0.5 points I claim full credit for this 
15) Are there two t tests ? Did the student correctly look up critical t values from the t table in the textbook or from
EXCEL functions?
16) The NULL and ALTERNATIVE hypotheses should be correctly stated for each test.
1.5 points I claim full credit for this 
17) Is there one F test? Did the student correctly look up critical values for F from the F table in the textbook? 1
point I claim full credit for this 
18) Are the t tests and F tests discussed somewhere in the text?
0.5 points I claim full credit for this 
19) You should mention possible future work to improve the model in the conclusion section.
Is there a punch line for conclusion? Usually this will comprise the gist of the following 3 things:
The X (spell out the name) variable has (has not) a statistically significant effect on Y (Spell out name).
The Z (spell out the name) variable has (has not) a statistically significant effect on Y (Spell out name).
The overall model is (is not) statistically significant. Note that possible future work is not called the punch line.
1
point I claim full credit for this 
Total Points earned in PART 3 are:
3 Out of 4.5
Grand Total of All points claimed are:
11 Out of 13. Auditor to state his itemized scores in red pen and sign.
Grand total of points given by the auditor are:
Out of 13