Week 8
John Hondros
September 21, 2015
Question One:
Briefly, what is Modigliani’s life-cycle theory of consumption? For an individual, draw someone’s consumption schedule over their life. Now draw their wealth over their life. At which periods are they saving and dissaving?
Modigliani’s life-cycle theory of consumtion stipulates that people make intelligent choices about how much they want to spend at different ages. A young person would have a higher rate of saving in order to accumulate wealth for their future retirement. This saving will continously increase untill a person is near retirement age. Once a person retires they begin to consume more and reduce their levels of investment.
What effects do population growth and economic growth have on savings rates?
A higher population growth rate will lead to more young people, meaning more people are saving rather than dissaving. this means that there will be a net-positive saving. If this is coupled with economic growth leading to a rise in incomes then this will compound the savings done by the young and lead to a higher savings rate.
In the Modigliani model, do consumers consume from current income?
In the modigliani model consumers do not consume from their current income but instead consume what is appropriate for their age. therefore their discrete income does not matter but their age does.
What is the impact in the model of an increase in transitory income?
If there is a transitory increase in income, in other words a brief increase that is not permanent, then the consumption in relation to income or the Marginal Propensity to Consume would not change. This is because consumers are factoring in the transitory nature of the income change and therefore saving the income increase they are gaining.
5. What are the implications of the theory for savings rates over the economic cycle?
The theory implies that the savings rate of a country is infact depedent on the economic growth of a country and not the absolute level that it has. Therefore rising incomes will increase the savings rate.
Question Two
Modelling
1. Plot consumption and GDP on the same plot
CY <- read.csv("dataweek8.csv")
CY <- CY %>%
mutate(logC = log(C),
logY = log(Y),
Clagged = lag(logC),
Ylagged = lag(logY))
# Variables added here were the Log of Consumption (C), the Log of GDP (Y) and the lagged term of those variables
CY <- CY %>%
mutate(Clagged.diff = c(NA, diff(Clagged)),
Ylagged.diff = c(NA, diff(Ylagged)),
logY.diff = c(NA, diff(logY)),
logC.diff = c(NA, diff(logC)))
# Variables added above are the differences of both contemperaneous and lagged log series
#Plotting the Log of C against the Log of Y
CY %>% ggplot(aes(x = logC, y = logY)) +
geom_point()
the variables do not seem to be stationary as they are ascending in a linear fashion, this may be a drift or a random walk.
2. Run a linear model for the logged values. What is the value of \(\beta_1\)? Is this a unit root series?
linmodC <- lm(logC ~ Clagged, data = CY, na.action="na.exclude")
summary(linmodC)##
## Call:
## lm(formula = logC ~ Clagged, data = CY, na.action = "na.exclude")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.0298837 -0.0115955 -0.0000182 0.0116133 0.0262341
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.129859 0.048578 2.673 0.01 *
## Clagged 0.992566 0.003806 260.816 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.0151 on 52 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9992, Adjusted R-squared: 0.9992
## F-statistic: 6.803e+04 on 1 and 52 DF, p-value: < 2.2e-16The value of \(\beta_1\) is 0.129859.
3. Run a forecasting linear model of the logged values. Extract the residuals from the model, add them as a new column to the data frame, and plot them over time. Do they appear to be stationary? If a residual is high in a given period, does this appear to indicate lower residuals in the future?
linlog <- lm(logC ~ Ylagged, data = CY, na.action = "na.exclude")
summary(linlog)##
## Call:
## lm(formula = logC ~ Ylagged, data = CY, na.action = "na.exclude")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.041780 -0.012940 0.000318 0.011284 0.042943
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.793746 0.069018 -11.5 6.69e-16 ***
## Ylagged 1.015045 0.005154 196.9 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01999 on 52 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.9987, Adjusted R-squared: 0.9986
## F-statistic: 3.879e+04 on 1 and 52 DF, p-value: < 2.2e-16CY <- CY %>%
mutate(residlog = resid(linlog))
CY %>% ggplot (aes(y=residlog, x=Date))+
geom_point()## Warning: Removed 1 rows containing missing values (geom_point).
4. Build a (linear) model for changes in log consumption. The left hand side of this model should be differenced log consumption at \(t\). On the right hand side should be lagged changes in log consumption (which capture “momentum”), and changes in log GDP at \(t\) (which captures a shift in GDP that could be affecting consumption). What are the coeffients? What interpretation do you give to them?
changeC <- lm(logC.diff ~ Clagged.diff + logY.diff, data = CY, na.action="na.exclude")
summary(changeC)##
## Call:
## lm(formula = logC.diff ~ Clagged.diff + logY.diff, data = CY,
## na.action = "na.exclude")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.029288 -0.007819 0.002662 0.009747 0.023384
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.010864 0.005625 1.931 0.0591 .
## Clagged.diff 0.238045 0.115436 2.062 0.0444 *
## logY.diff 0.463235 0.102777 4.507 3.98e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01293 on 50 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.334, Adjusted R-squared: 0.3074
## F-statistic: 12.54 on 2 and 50 DF, p-value: 3.863e-05The coefficients are: (Intercept) 0.010864 Clagged.diff 0.238045 logY.diff 0.463235
These coefficients may show that the differences in Consumption (logC.diff) are positively affected by the “momentum” of consumption and the shift in GDP that has affected consumption.
5. Build the same model again, but this time including the lagged residuals from the the model in step 3. What is the coefficient on this term? What interpretation do you give to it?
changeClagged <- lm(logC.diff ~ Clagged.diff + logY.diff + residlog, data = CY, na.action="na.exclude")
summary(changeClagged)##
## Call:
## lm(formula = logC.diff ~ Clagged.diff + logY.diff + residlog,
## data = CY, na.action = "na.exclude")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.029662 -0.006467 0.002717 0.009186 0.022209
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.011131 0.005683 1.959 0.055869 .
## Clagged.diff 0.244976 0.116881 2.096 0.041276 *
## logY.diff 0.448214 0.106867 4.194 0.000114 ***
## residlog 0.052829 0.093795 0.563 0.575838
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01302 on 49 degrees of freedom
## (2 observations deleted due to missingness)
## Multiple R-squared: 0.3383, Adjusted R-squared: 0.2978
## F-statistic: 8.35 on 3 and 49 DF, p-value: 0.0001382The variables are:
(Intercept) 0.011131
Clagged.diff 0.244976
logY.diff 0.448214
residlog 0.052829
6. What is the Marginal Propensity to Consume?
MPCmod <- lm(logY.diff ~ logC.diff, data = CY, na.action="na.exclude" )
summary(MPCmod)##
## Call:
## lm(formula = logY.diff ~ logC.diff, data = CY, na.action = "na.exclude")
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.050329 -0.007990 -0.000444 0.007803 0.032545
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.013542 0.005033 2.691 0.00956 **
## logC.diff 0.595070 0.131536 4.524 3.55e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.01484 on 52 degrees of freedom
## (1 observation deleted due to missingness)
## Multiple R-squared: 0.2824, Adjusted R-squared: 0.2686
## F-statistic: 20.47 on 1 and 52 DF, p-value: 3.551e-05MPC = \(\Delta C / \Delta Y\)
MPC = slope of Consumption against Income
MPC = logC.diff = 0.595070