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MSIA 402 - Homework 2 - 2016-10-05.Rmd
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MSIA 402 - Homework 2 - 2016-10-05.Rmd
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---
title: "MSIA410 - Homework 2 - 2016-10-05"
author: "Jeff Parker"
date: "October 5, 2016"
output: html_document
---
## 3.1
y = Beta * x + Epsilon
$$\left[\begin{array}
{rrr}
y_1 \\
y_2 \\
y_n
\end{array}\right]
$$
## 3.6
n = 26
p = 5
q = 3
r2_p = .9
r2_q = .8
F = ((r2_p - r2_q)/(p - q))/((1-r2_p)/(n - (p + 1)))
## 3.12
```{r setup, include=FALSE}
#a)
x <- c(1,1,1,2,1,3,1,4,1,5)
y <- c(2,6,7,9,10)
X_Bold <- matrix(x, nrow = 5, ncol = 2, byrow = TRUE)
Y_Bold <- matrix(y, nrow = 5, ncol = 1)
#b)
X_Transpose <- t(X_Bold)
X_Transpose_X_Bold <- X_Transpose %*% X_Bold
inverse <- solve(X_Transpose_X_Bold) #Gets the inverse
X_Transpose_X_Bold %*% inverse #Checking that this gets an identity matrix
#c)
X_Transpose_Y <- X_Transpose %*% Y_Bold
#d)
B_Hat <- inverse %*% X_Transpose_Y
```
## 3.13
a) Fit Cobb-Douglas production function
```{r setup, include=FALSE}
Cobb.Douglas <- read.csv("C:/Users/Jeff/Downloads/Cobb-Douglas.csv")
lmfit <- lm(log(Cobb.Douglas$output) ~ log(Cobb.Douglas$capital)+log(Cobb.Douglas$labor))
summary(lmfit)
```
Intercept: Beta_0 = e^-1.71146
Capital: Beta_1 = 0.20757
Labor: Beta_2 = 0.71485
b) Check to see if there are constant returns to scale
i) t-test
H_0 : Beta_1 + Beta_2 - 1 = 0
```{r setup, include=FALSE}
X_Bold <- matrix(c(Cobb.Douglas$capital,Cobb.Douglas$labor), nrow = 569, ncol = 2)
var_beta_1 <- var(log(Cobb.Douglas$capital))
var_beta_2 <- var(log(Cobb.Douglas$labor))
cov_beta1_beta2 <- cov(log(Cobb.Douglas$capital),log(Cobb.Douglas$labor), use = 'everything', method = 'pearson')
t = ((var_beta_1+var_beta_2)-1)/(sqrt(var_beta_1+var_beta_2+2*cov_beta1_beta2))
df = nrow(Cobb.Douglas) - ncol(Cobb.Douglas)
dt(t,df)
drop1(lmfit, test = "F")
```
## 3.14
```{r setup, include=FALSE}
Region <- c(1,2,3,4,5,6,7,8,9,10)
Salesmen <- c(31,46,40,49,38,49,31,38,33,42)
Expenditure <- c(1.85,2.80,2.20,2.85,1.80,2.80,1.85,2.30,1.60,2.15)
Sales <- c(4.20,7.28,5.60,8.12,5.46,7.42,3.36,5.88,4.62,5.88)
Sales_data <- data.frame(Region,Salesmen,Expenditure,Sales)
#a)
sm_std <- (Salesmen - mean(Salesmen))/sd(Salesmen)
exp_std <- (Expenditure - mean(Expenditure))/sd(Expenditure)
sales_std <- (Sales - mean(Sales))/sd(Sales)
X_Bold <- matrix(c(sm_std,exp_std), nrow=10,ncol=2)
Y_Bold <- matrix(sales_std, nrow=10, ncol=1)
g <- (1/(nrow(Sales_data)-1))
t <- t(X_Bold)
R_Bold <- g * t(X_Bold) %*% X_Bold
r_bold <- g * t(X_Bold) %*% Y_Bold
#b)
Beta_Hat_Bold <- solve(R_Bold) %*% r_bold
#c)
lmfit <- lm(Sales ~ Salesmen + Expenditure)
sd_Sales <- sd(Sales)
sd_Salesmen <- sd(Salesmen)
sd_Exp <- sd(Expenditure)
#Beta_Hat_Salesmen = 0.1922
#Beta_Hat_Expenditure = 0.3406
#Checking Beta_Hat_Salesment matches the scaled Beta_Star_Hat
#Beta_Hat_Salesmen = 0.1922
#The equation below should equal the amount above
Beta_Hat_Bold[1]*sd_Sales/sd_Salesmen
#Beta_Hat_Expenditure = 0.3406
#The equation below should equal the amount above
Beta_Hat_Bold[2]*sd_Sales/sd_Exp
#Beta_Hat_Salesmen = 0.1922
#Beta_Hat_Expenditure = 0.3406
#Beta_Hat_Star_Salesmen = 0.8752107
#Beta_Hat_Star_Expenditure = 0.1047290
#In this case the scaled predictor is more inititive to used because the variables have different scales. Looking just at the Beta's one might think that expenditure has more effect on sales when in fact it is just a percent change in the expenditure that has that effect. The number of salesmen have a much higher effect in sales.
```
## 3.14
```{r setup, include=FALSE}
Salary <- read.csv("C:/Users/Jeff/Downloads/TEMCO.csv")
Salary$Female <- as.integer(ifelse(Salary$Gender == 'Female',1,0))
Salary$Advert <- as.integer(ifelse(Salary$Dept == 'Advertse',1,0))
Salary$Engg <- as.integer(ifelse(Salary$Dept == 'Engineer',1,0))
Salary$Sales <- as.integer(ifelse(Salary$Dept == 'Sales',1,0))
#Just taking the natural log of salary
lmfit <- lm(log(as.numeric(Salary$Salary)) ~ Salary$YrsEm + Salary$PriorYr + Salary$Education + Salary$Super + Salary$Female + Salary$Advert + Salary$Engg + Salary$Sales)
summary(lmfit)
#Taking the natural log of everything
Salary <-data.frame(Salary$Salary,Salary$YrsEm+1,Salary$PriorYr+1,Salary$Education+1,Salary$Super+1,Salary$Female+1,Salary$Advert+1,Salary$Engg+1,Salary$Sales+1)
lmfit <- lm(log(as.numeric(Salary$Salary)) ~ log(as.numeric(Salary$YrsEm)) + log(as.numeric(Salary$PriorYr)) + log(as.numeric(Salary$Education)) + log(as.numeric(Salary$Super)) + log(as.numeric(Salary$Female)) + log(as.numeric(Salary$Advert)) + log(as.numeric(Salary$Engg)) + log(as.numeric(Salary$Sales)), )
summary(lmfit)
```