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

SFX_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/Stocks_FX_1973.csv",head=TRUE,sep=",")
names(SFX_da)
x_sp <- SFX_da$SP500
x_ibm <- SFX_da$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])
lr_sp <- log(x_sp[-1]/x_sp[-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

####  Testing Linear Hypothesis in the Fama-French 3 factor Linear Model

# OLS results
fit_ibm_ff3 <- lm(ibm_x ~ Mkt_RF + SMB + HML)
summary(fit_ibm_ff3)


## Test Linear Restrictions using R package "car"
library(car)

linearHypothesis(fit_ibm_ff3, c("SMB = 0.2","HML = 0.6"), test="F")	# Exact F test
linearHypothesis(fit_ibm_ff3, c("SMB = 0.2","HML = 0.6"), test="Chisq")	# Asymptotic F-test

linearHypothesis(fit_ibm_ff3, c("SMB = HML"), test="F")	# testing beta_SMB = beta_HMB

linearHypothesis(fit_ibm_ff3, c("SMB + HML = 1"), test="F")	# testing beta_SMB + beta_HMB = 1



####  Testing with Unrestricted and Restricted Estimations 

## with package lmtest
library(lmtest)
fit_wU <- lm (ibm_x ~ Mkt_RF + SMB + HML)
fit_wR <- lm (ibm_x ~ Mkt_RF)
waldtest(fit_wU, fit_wR)


##### Testing for Structural Change (Chow Test): Dot.com Bubble (T_sb = 347)

## Compute Chow test with R package strucchange
T_sb <- 347
library(strucchange)
sctest(ibm_x ~ Mkt_RF + SMB + HML, type = "Chow", point = T_sb)



#####  Testing Non-Nested Models

### Encompassing Test
SF_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/SpFor_prices.csv", head=TRUE, sep=",")
x_date <- SF_da$Date
x_S <- SF_da$GBPSP				# Extract USD/GBP exchange rate
x_F3m <- SF_da$GBP3M
i_us3 <- SF_da$Dep_USD3M			# Extract US 3-mo interest rate (i_d)
i_uk3 <- SF_da$Dep_UKP3M			# Extract UK 3-mo interest rate  (i_f_
cpi_uk <- SF_da$UK_CPI				# Extract UK CPI 
cpi_us <- SF_da$US_CPI				# Extract US CPI 
T <- length(x_S)
int_dif <- (i_us3[-1] - i_uk3[-1])/100		# (i_d - i_f): interest rate differential
lr_usdgbp <- log(x_S[-1]/x_S[-T])		# log changes in USD/GBP exchange rate
I_us <- log(cpi_us[-1]/cpi_us[-T])		# US Inflation rate (I_d): log changes in US CPI 
I_uk <- log(cpi_uk[-1]/cpi_uk[-T])		# UK Inflation rate (I_f): log changes in UK CPI 
inf_dif <- (I_us - I_uk)			# (I_d - I_f): inflation rate differential

fit_me <- lm(lr_usdgbp ~ int_dif + inf_dif)		# Encompassing Model (ME)
summary(fit_me)

## With R package lmtest
library(lmtest)
fit_m1 <- lm(lr_usdgbp ~ int_dif)			# Restricted Model 1
fit_m2 <- lm(lr_usdgbp ~ inf_dif) 			# Restricted Model 2
encomptest(fit_m1, fit_m2)


### J-Test
y <- lr_usdgbp

## Model 1
fit_m1 <- lm( y ~ int_dif)
summary(fit_m1)

## Fitted Values of Model 1 into Model 2
y_hat1  <- fitted(fit_m1)
fit_J1 <- lm( y ~ inf_dif + y_hat1)
summary(fit_J1)


## Model 2
fit_m2 <- lm( y ~ inf_dif)
summary(fit_m2)

## Fitted Values of Model 2 into Model 1
y_hat2  <- fitted(fit_m2)
fit_J2 <- lm( y ~ int_dif + y_hat2)
summary(fit_J2)


## Using R package lmtest
library(lmtest)
fit_m1 <- lm(lr_usdgbp ~ int_dif)
fit_m2 <- lm(lr_usdgbp ~ inf_dif)
jtest(fit_m1, fit_m2)


#####  Omitted Variables
Fger_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/FX_USA_GER.csv", head=TRUE, sep=",")
us_CPI <- Fger_da$US_CPI				# Extract US CPI 
us_M1 <- Fger_da$US_M1					# Extract US M1 (Money Supply)
us_i <- Fger_da$US_I3M					# Extract US interest rates (in %)
us_GDP <- Fger_da$US_GDP				# Extract US GDP 
us_u <- Fger_da$US_UN					# Extract US unemployment rate  
S_ger <- Fger_da$USD_EUR				# Extract USD/EUR exchange rate 

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")