############# Ommited Variables, Trilogy of Tests (Wald, LR & LM tests) & Specification tests (LM Tests & RESET) (Code #8) #############


#####  Omitted Variables
Fger_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/FX_USA_GER.csv", head=TRUE, sep=",")
us_CPI <- Fger_da$US_CPI
us_M1 <- Fger_da$US_M1
us_i <- Fger_da$US_I3M
us_GDP <- Fger_da$US_GDP
ger_CPI <- Fger_da$GER_CPI
us_u <- Fger_da$US_UN
S_ger <- Fger_da$USD_EUR

T <- length(us_CPI)
us_I <- log(us_CPI[-1]/us_CPI[-T])			# US Inflation: (Log) Changes in CPI 
us_mg <- log(us_M1[-1]/us_M1[-T]) 			# US Money Growth: (Log) Changes in M1
e_ger <- log(S_ger[-1]/S_ger[-T]) 			#  (Log) Changes in USD/EUR
us_i_1 <- us_i[-1]					# Adjust sample size of untransformed data 
us_u_1 <- us_u[-1]					# Adjust sample size of untransformed data 
us_i_0 <- us_i[-T]					# lagged interest rates, by removing T observation
xx_i <- cbind(us_I ,e_ger, us_mg, us_u_1)		# X matrix


fit_i <- lm(us_i_1 ~ xx_i)
summary(fit_i)

xx_i <- cbind(us_I ,e_ger, us_mg, us_u_1, us_i_0)	# X matrix with lagged interest rates
fit_i <- lm(us_i_1 ~ xx_i)
summary(fit_i)



##### Trilogy of Tests
SFX_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/Stocks_FX_1973.csv",head=TRUE,sep=",")
x_ibm <- SFX_da$IBM
x_Mkt_RF<- SFX_da$Mkt_RF
x_SMB <- SFX_da$SMB
x_HML <- SFX_da$HML
x_RF <- SFX_da$RF

T <- length(x_ibm)
lr_ibm <- log(x_ibm[-1]/x_ibm[-T])
x0 <- matrix(1,T-1,1)
 Mkt_RF <- x_Mkt_RF[-1]/100
 SMB <- x_SMB[-1]/100
 HML <- x_HML[-1]/100
 RF <- x_RF[-1]/100
 ibm_x <- lr_ibm - RF						# IBM excess returns 


### Specification Test: Using Wald & LM Tests to test for Omitted Variables (SMB + HML in CAPM)
library(lmtest)
fit_ibm_ff3 <- lm (ibm_x ~ Mkt_RF + SMB + HML)
fit_ibm_capm <- lm (ibm_x ~ Mkt_RF)

waldtest(fit_ibm_ff3, fit_ibm_capm)

lrtest(fit_ibm_ff3, fit_ibm_capm)


### Specification Test: Using LM Tests to test for Omitted Variables (SMB + HML in CAPM)

T <- length(ibm_x)
resid_r <- fit_ibm_capm$residuals				# extract residuals from R model
fit_lm <- lm (resid_r ~ Mkt_RF + SMB + HML)			# auxiliary regression
summary(fit_lm)
R2_r <- summary(fit_lm)$r.squared			 	# extract  R2 from fit_lm 
R2_r

LM_test <- R2_r * T
LM_test

qchisq(.95, df = 2) 	# chi-squared (df=2) value at 5% level

p_val <- 1 - pchisq(LM_test, df = 2)  	# p-value of LM_test 
p_val


### Specification Test: Using LM Tests to test for Non linearities (Omitted Variable are Mkt_RF^2, SMB^2 + HML^2)

## Wald Test for including in FF3 Model only MKt_RF^2 
Mkt_RF2 <- Mkt_RF^2 
fit_ibm_ff3_2 <- lm (ibm_x ~ Mkt_RF + SMB + HML + Mkt_RF2)
summary(fit_ibm_ff3_2)

## LM Test for including in FF3 Model MKt_RF^2, SMB^2 + HML^2 
SMB2 <- SMB^2
HML2 <- HML^2

resid_r <- fit_ibm_ff3$residuals
fit_lm <- lm (resid_r ~ Mkt_RF + SMB + HML + Mkt_RF2 + SMB2 + HML2)

R2_r <- summary(fit_lm)$r.squared
LM_test <- R2_r * T
LM_test

qchisq(.95, df = 3) 				# 95% quantile for Chi-square distribution
p_val <- 1 - pchisq(LM_test, df = 3)  			# p-value of LM_test 
p_val



### Specification Test: RESET Test
y_hat <- fitted(fit_ibm_ff3)
y_hat2 <- y_hat^2
fit_ramsey <- lm(ibm_x ~ Mkt_RF + SMB + HML + y_hat2)
summary(fit_ramsey)

y_hat4 <- y_hat^4
fit_ramsey2 <- lm(ibm_x ~ Mkt_RF + SMB + HML + y_hat4)
summary(fit_ramsey2)

fit_ramsey2 <- lm(ibm_x ~ Mkt_RF + SMB + HML + y_hat2 + y_hat4)
summary(fit_ramsey2)


## Using R package lmtest
library(lmtest)
resettest(fit_ibm_ff3, power=2, type="fitted")

resettest(fit_ibm_ff3, power=4, type="fitted")