Sample Essay On SR Numeric Aggregate Personal Savings

Assignment 1: First Regression

(a) Describe the economic issue.
In order to explain an individual's consumption patterns based on their stage in life and the resources available to them over their lifetime, Modigliani proposed the Life-Cycle Hypothesis (Modigliani 160-217; Japelli and Modigliani 7), which states that the savings ratio of an individual could be explained by per-capita disposable income (dpi), the percentage rate of change in per-capita disposable income, the percentage of population less than 15 years old and the percentage of the population over 75 years old.
(b) Describe the data
The LifeCycleSavings data included in R contains data on the savings ratio between 1960 and 1970. These data were obtained from Belsley, Kuh and Welsch (1980), who in turn obtained the data from Sterling (1977). This is a dataset of 50 observations and 5 variables:
variable class description

pop15 numeric % of population under 15
pop75 numeric % of population over 75
dpi numeric real per-capita disposable income
ddpi numeric % growth rate of dpi
(c) Describe your model

The multiple linear regression model performed in R using the LifeCycleSavings data is as follows:

model1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, data = data)
This model predicts the value of aggregate personal savings (outcome variable) according to the percentage of population under 15, the percentage of population over 75, the real per-capita disposable income and the percentage growth rate of dpi (explanatory variables).
(d) State your hypotheses in terms of the model

For the Wald test (to test if any of the coefficients are equal to zero or not):

H0: βpop15 = βpop75 = βpop75 = βddpi= 0

H1: βpop15 or βpop75 or βpop75 or βddpi ≠ 0

Where β is the coefficient for each variable. If a coefficient is equal to zero, there is no linear relationship with the outcome variable (Altman 336-340).

The F-test references to the explained and unexplained variance, and the hypotheses are (Altman 197):

H0: explained and unexplained variances in the model are equal
H1: explained and unexplained variances in the model are not equal
(e) Present the regression results:

Call:

lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = data)

Residuals:

Min 1Q Median 3Q Max
-8.3318 -2.5187 -0.4661 2.2810 9.7113

Coefficients:

Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.869e+01 7.916e+00 3.624 0.000793 ***
pop15 -4.600e-01 1.549e-01 -2.971 0.004950 **
pop75 -1.881e+00 1.230e+00 -1.529 0.133865
dpi -1.566e-05 9.713e-04 -0.016 0.987212
ddpi 3.954e-01 2.010e-01 1.967 0.055950 .
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.818 on 41 degrees of freedom

Multiple R-squared: 0.3387, Adjusted R-squared: 0.2742
F-statistic: 5.25 on 4 and 41 DF, p-value: 0.001652
(f) Interpret the regression results:
When all predictors are zero, the personal savings are 2.869e+01. For each 1% increase in the population under 15, personal savings decrease by -4.600e-01 units. For each 1% increase in the population over 75, personal savings decrease by -1.881e+00 units. For every unit increase in the dpi, personal savings decrease by -1.566e-05 units, while for each 1% increase in the growth rate of dpi, personal savings increase by 3.954e-01 units. This model explains 33.87% of the variability (27.42% when adjusted by the number of predictors).
(g) State whether your hypotheses were supported by the data
Only the intercept, and the pop15 variables were significantly different from zero, which means that at a 5% significance level pop75, dpi and ddpi do not add any significant information to the model. However, it is important to take into account that the 5% significant level is arbitrary, so this might not be economically significant in real life. Furthermore, the F-statistic is significant (p value 0.001652) which means that there is a linear relationship between the predictors and the outcome variable (Altman 197).
(h) Draw conclusions
In the presented model, the percentage growth rate of the dpi increases the savings, while the percentage of people under 15, the percentage of people over 75 and the real per-capita disposable income actually decrease the savings. Although this model explains about one third of the variability of personal savings based on the population under 15 and over 75, the dpi and the growth rate of the dpi, it is recommended to run model selection procedures (e.g. AIC, BIC) to explore if there is a more parsimonious model.
Graphs

Works Cited

Altman, Douglas G. Practical statistics for medical research. CRC Press, 1990.
Belsley, D.A., Kuh, E., and Welsch, R. E.. Regression Diagnostics. New York: Wiley, 1980. Print.
Jappelli, Tullio, and Modigliani, Franco. "The age-saving profile and the life-cycle hypothesis." Long-run Growth and Short-run Stabilization: Essays in Memory of Albert Ando (1998): 12.
Modigliani, Franco. "The life cycle hypothesis of saving, the demand for wealth and the supply of capital." Social Research (1966): 160-217.
Sterling, Arnie (1977) Unpublished BS Thesis. Massachusetts Institute of Technology.