############# Chow Test and Non-Nested Tests, Trilogy of Tests (Wald, LR & LM tests) & Specification tests (LM Tests & RESET) (Code #8 - Lecture 13) #############

SFX_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/Stocks_FX_1973.csv",head=TRUE,sep=",")
names(SFX_da)
x_date <- SFX_da$Date
x_sp <- SFX_da$SP500				# extract S&P500 price data
x_ibm <- SFX_da$IBM				# extract IBM price data
x_ko <- SFX_da$KO				# extract Coca-Cola price data
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])
lr_ko <- log(x_ko[-1]/x_ko[-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
ko_x <- lr_ko - RF				# IBM excess returns

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

# OLS results
y <- ibm_x
x <- cbind(Mkt_RF, SMB, HML)
T <- length(y)


fit_y_ff3 <- lm(y ~ Mkt_RF + SMB + HML)
summary(fit_y_ff3)

####  Testing with Unrestricted and Restricted Estimations 

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



##### Testing for Structural Change (Chow Test): Dot.com Bubble (T_sb = 347)
## Pooled RSS  (1973 - 2024)
e_p <- fit_y_ff3$residuals			# extract residuals from pooled FF regression
RSS_p <- sum(e_p^2)

## Before T_sb RSS  (1973 - 2001)
T_sb <- 347
x_date[T_sb]
y1 <- y[1:T_sb]
x1 <- x[1:T_sb,] 
fit_y_ff3_1 <- lm(y1 ~ x1 -1)			# Add -1 inside lm() so no constant is added to regressors
summary(fit_y_ff3_1)
e_1 <- fit_y_ff3_1$residuals			# extract residuals from pooled FF regression
RSS_1 <- sum(e_1^2)

## After T_sb RSS  (2002 - 2024)
T2 <- T_sb + 1
y2 <- y[T2:T]
x2 <- x[T2:T,] 
fit_y_ff3_2 <- lm(y2 ~ x2 -1)			# Add -1 inside lm() so no constant is added to regressors
summary(fit_y_ff3_2)
e_2 <- fit_y_ff3_2$residuals			# extract residuals from pooled FF regression
RSS_2 <- sum(e_2^2)

## Compute Chow test
k <- length(fit_y_ff3$coefficients)
F_chow_t <- ((RSS_p - (RSS_1 + RSS_2))/k) / ((RSS_1 + RSS_2)/(T-2*k))		# Chow test (F-test) 
F_chow_t
p_val <- 1 - pf(F_chow_t, df1=k, df2=(T - 2*k)) 	# p-value(F_t) under e normal
p_val


## Compute Chow test with R package strucchange (need to install package sandwich too)
T_sb <- 347
library(strucchange)
library(sandwich)
sctest(ibm_x ~ Mkt_RF + SMB + HML, type = "Chow", point = T_sb)
sctest(ko_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
## Model for US 3-mo interest rates as a function of US Inflation, Exchange Rates, and Canadian 3-mo int rate
Fca_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/FX_USA_CAN.csv", head=TRUE, sep=",")
names(Fca_da)
summary(Fca_da)
us_CPI <- Fca_da$US_CPI					# Extract US CPI 
us_U <- Fca_da$US_U					# Extract US Unemployment rate
us_i <- Fca_da$US_I3M					# Extract US interest rates (in %)
can_i <- Fca_da$Can_I3M					# Extract Canadian interest rate
can_CPI <- Fca_da$Can_CPI				# Extract US CPI 
S_cad <- Fca_da$CAD_USD					# Extract USD/EUR exchange rate 
x_Mkt_RF<- Fca_da$Mkt_RF
x_SMB <- Fca_da$SMB
x_HML <- Fca_da$HML


T <- length(us_CPI)
us_I <- log(us_CPI[-1]/us_CPI[-T])			# US Inflation: (Log) Changes in CPI 
can_I <- log(can_CPI[-1]/can_CPI[-T]) 			# US Money Growth: (Log) Changes in M1
e_cad <- log(S_cad[-1]/S_cad[-T]) 			# (Log) Changes in USD/EUR
us_i_1 <- us_i[-1]					# Adjust sample size of untransformed data 
can_i_1 <- can_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
SMB <- x_SMB[-1]/100
HML <- x_HML[-1]/100
xx_i <- cbind(us_I ,e_cad, can_i_1)		# X matrix
summary(xx_i)

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

xx_i <- cbind(us_I ,e_cad, can_i_1, us_i_0)	# X matrix with lagged interest rates
fit_i <- lm(us_i_1 ~ xx_i)
summary(fit_i)


## Irrelevant Variables (Add SMB as an explanatory variable)
xx_i <- cbind(us_I ,e_cad, can_i_1, us_i_0, SMB)	# 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