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Assignment 1: First Regression

(a) Describe the economic issue.
Modigliani (Modigliani 160-217) tried to explain consumption patterns, using the life-cycle hypothesis, based on individuals’ stage in life and the resources available to them over a lifetime. Within this theory, it is discussed that the savings ratio can be described by age, the percentage of the population is under 15 years old and the percentage of the population is over 75 years old, disposable income per-capita and the change in percentage rate in disposable income per-capita. Where the savings ratio is the aggregate personal saving divided by disposable income.
(b) Describe the data
The LifeCycleSavings dataset has savings ratio information (1960-1970). These data were collected from Belsley, Kuh and Welsch (1980), and they got them in turn from Sterling (1977). It has 5 variables and 46 observations. The variables are sr (aggregate personal savings), pop15 (percentage of population under 15), pop75 (percentage of population over 75), dpi (real per-capita disposable income) and ddpi.
(c) Describe your model
According to the life cycle hypothesis, savings ratio can be defined by disposable income per-capita and the change in percentage rate in disposable income per-capita and age. So in this linear model we can construct the dependent variable sr by Y , pop15 by X1, pop75 by X2, dpi by X3 and ddpi by X4. Finally, the linear relation can be done as Y = b0+ b1 X1 + b2 X2+ b3 X3+ b4 X4, where b is the constant and can be estimated by using the method of least squares.
(d) State your hypotheses in terms of the model
There are two tests when performing a regression model. The Wald test evaluates whether the predictors are equal to zero or not: The null hypothesis (H0) is: βpop15 = βpop75 = βpop75 = βddpi= 0; while the alternative hypothesis (H1) is: βpop15 or βpop75 or βpop75 or βddpi ≠ 0, where β represents the variable’s coefficient. There is no linear relationship between the predictor and the outcome variable if the coefficient equals zero.
The F-test evaluates if the explained and unexplained variances are equal or not: The null hypothesis (H0) is: unexplained and explained variances in the model are statistically equal; the alternative hypothesis (H1) is: unexplained and explained variances are not statistically equal.
(e) Present the regression results:

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

Residuals:

Min 1Q Median 3Q Max
-8.0272 -2.4863 -0.1111 2.3598 10.0434

Coefficients:

Estimate Std. Error t value Pr(>|t|)
(Intercept) 30.2004668 7.4349000 4.062 0.000214 ***
pop15 -0.4988306 0.1477336 -3.377 0.001617 **
pop75 -1.7804309 1.0891951 -1.635 0.109783
dpi -0.0005809 0.0009309 -0.624 0.536053
ddpi 0.3824217 0.1970547 1.941 0.059193 .
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Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.768 on 41 degrees of freedom

Multiple R-squared: 0.3554, Adjusted R-squared: 0.2925
F-statistic: 5.652 on 4 and 41 DF, p-value: 0.00102
(f) Interpret the regression results:
The value of the F-statistic is 5.652 with a p-value 0.00102. The adjusted R-squared value of the model is 0.2925 means that the model explains 29.25% of the outcome variation. The coefficient of the variable pop15 is -0.4988306, which means that the aggregate personal savings is decreased by 0.4988306 units with every unit increase of pop15. Similarly, the coefficient of the variable pop75 is -1.7804309, and means that with every unit increase of this variable the aggregate personal savings is decreased by 1.7804309 units. A unit increase in dpi makes aggregate personal savings decrease by 0.0005809 units. On the contrary, a unit increase in ddpi makes the outcome variable increase by 0.3824217 units.
(g) State whether your hypotheses were supported by the data
Further from the above regression output, the p-value of the coefficient of the variable pop15 is 0.001617 and the one for ddpi is 0.059193, and both are lower than 0.10. the p-value of the coefficient of the variable pop75 is 0.109783, the one for the variable dpi is 0.536053, and both are greater than 0.10, thus statistically significant at a 10% level. Therefore only pop15 and ddpi contribute significantly to the model at this level. Furthermore, the F-statistic is significant (p value 0.00102) which means that the predictors and the outcome variable are linearly related.
(h) Rehypothesis
A second model including only those variables significant for the first model was elaborated. The new hypotheses for Wald test of this model are: H0: βpop15 = βddpi= 0 and H1: βpop15 or βddpi ≠ 0. This also means that not all variables initially considered successfully predict aggregate savings over lifetime, but only population under 15 and the percentage of growth rate of the dpi. This new model explains 25.37% of the variance (adjusted R-squared), with both predictors being statistically significant at a 10% level (p-values 0.00139 for pop15 and 0.03541 for ddpi). The F-statistic was also significant (p-value 0.0006977). Therefore, although this model explains less variance, all predictors are statistically significant. Furthermore, for each unit increase of pop15 and ddpi, sr decreases by 0.22109 units and increases by 0.42390 units, respectively.
(i) Draw conclusions
In the first model, the percentage growth rate of the dpi increases the savings, but the percentage of people over 75, the percentage of people under 15 and the real per-capita disposable income decrease them. However, not all predictors were statistically significant. In the second model, the direct and inverse relationships between ddpi and pop15 are still present, and both predictors were statistically significant at a 10% level, but less variance is explained.

Graphs (first model)

Graphs (second model)

Works Cited

Belsley, D.A., Kuh, E., and Welsch, R. E.. Regression Diagnostics. New York: Wiley, 1980. Print.
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.