(9-1) GARCH model

Nao Mimoto - Dept. of Statistics : The University of Akron

1. Stylized Facts of Financial Return

Financial (log) Return

\[\begin{align*} Y_t &= \mbox{ Stock Price (observation) } \\ \\ X_t &= \ln(Y_t) - \ln(Y_{t-1}) \hspace{5mm} : \mbox{ log-return } \end{align*}\]

__Ex: Apple

  library(quantmod)
  source('https://nmimoto.github.io/R/TS-00.txt') 

  getSymbols("AAPL")    #- download from Yahoo!

## [1] "AAPL"

  Apple <- Ad(AAPL)['2017::2019']

  plot(Apple)

  plot(log(Apple))

  plot(diff(log(Apple)))

  hist(diff(log(Apple)), 20)

Stylized Facts

Not normal - Heavy Tailed unconditional and conditional distribution
Uncorrelated
Squares are correlated
Clustering
Asymmetry

  Randomness.tests(as.numeric(diff(log(Apple))))

##   B-L test H0: the series is uncorrelated
##   M-L test H0: the square of the series is uncorrelated
##   J-B test H0: the series came from Normal distribution
##   SD         : Standard Deviation of the series

##       BL15 BL20  BL25 ML15 ML20 JB    SD
## [1,] 0.189 0.15 0.226    0    0  0 0.016

__Ex: SP500

  library(quantmod)
  getSymbols("^GSPC")    #- SP500 download from Yahoo!

## [1] "^GSPC"

  SP500 <- Ad(GSPC)['2000::']

  plot(SP500, type="l")

  plot(log(SP500), type="l")

  plot(diff(log(SP500)), type="l")

  Randomness.tests(as.numeric(diff(log(SP500))))

##   B-L test H0: the series is uncorrelated
##   M-L test H0: the square of the series is uncorrelated
##   J-B test H0: the series came from Normal distribution
##   SD         : Standard Deviation of the series

##      BL15 BL20 BL25 ML15 ML20 JB    SD
## [1,]    0    0    0    0    0  0 0.013

We need a model to capture these characteristics.

2. ARCH model

Engle (1985) AutoRegressive Conditionally Heteroscedastic Model
Won Nobel Prize in Economics

\[\begin{align*} Y_t &= \sigma_t e_t \hspace{10mm} e_t \sim_{iid} N(0,1) \\\\ \sigma_t^2 &= \omega+ \alpha Y_{t-1}^2 \end{align*}\]
(unconditional) Mean \[ E(Y_t) = \sigma_t E(e_t) = 0 \]
(unconditional) Variance \[ V(Y_t) = \frac{\omega}{1-\alpha} \hspace{5mm} 0<\alpha<1 \]

Cond’l Mean and Var of ARCH

Conditional mean \[ E[Y_t \Big| Y_{t-1}, e_{t-1}, \ldots] \hspace{3mm} = \hspace{3mm} \sigma_t E[e_t] = 0 \]
Conditional variance \[ V[Y_t \Big| Y_{t-1}, e_{t-1}, \ldots] \hspace{3mm} = \hspace{3mm} \sigma_t^2 V[e_t] = \sigma_t^2 \]

ARCH is uncorrelated

\(Y_t\) is uncorrelated, so it will pass the Ljung-Box test.
But \(Y_t^2\) is correlated, so it will NOT pass the McLeod-Li test.

Cond’l Mean and Var of ARMA

Suppose \(Y_t\) is AR(1) series observation
(unconditional) Mean \[ E( Y_t ) = 0 \]
(unconditional) Variance \[ V( Y_t ) = \gamma(0) = (1+\phi_1^2) \sigma^2 \]

Conditional Mean

Conditonal Mean:
\[ E\Big(Y_t \hspace{2mm} \Big| \hspace{2mm} \mbox{all variables realized by yesterday. }\Big) \]
Conditonal mean of AR(1): \(Y_t = \phi Y_{t-1} + e_t\) \[\begin{align*} E\Big(Y_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= E \Big(\phi Y_{t-1} + e_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ \\ &= \phi Y_{t-1} + E(e_t) \hspace{2mm} = \hspace{2mm} \phi Y_{t-1}. \end{align*}\]
Conditonal variance \[\begin{align*} V\Big(Y_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= V\Big(\phi Y_{t-1} + e_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ \\ &= V(e_t) \hspace{2mm} = \hspace{2mm} \sigma^2 \end{align*}\]

    n=200
    Y <- arima.sim(n = n, list(ar = c(0.8) ))
    
    v.AR1 = (1+.8^2)*1
    
    plot(Y, type="o", main="AR(1)", xlim=c(0,n*1.1))
    abline(h=0)
    abline(h=c(1,-1)*1.96*v.AR1, col="red")

    lines(n+1, .8*Y[n], type="p", col="blue")
    lines(c(n+1,n+1), .8*Y[n]+c(1.96,-1.96), type="p", col="red")

Suppose \(Y_t\) is MA(1) series observation
(unconditional) Mean \[ E( Y_t ) = 0 \]
(unconditional) Variance \[ V( Y_t ) = \gamma(0) = (1+\theta_1^2) \sigma^2 \]
Conditonal mean of MA(1): \(Y_t = e_t + \theta_1 e_{t-1}\) \[\begin{align*} E\Big(Y_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) &= E \Big( e_t + \theta_1 e_{t-1} \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) \\\\ &= E ( e_t )+ \theta_1 e_{t-1} \hspace{2mm} = \hspace{2mm} \theta_1 e_{t-1} \end{align*}\]

Note that \(e_{t-1}\) is not observable.
Conditonal variance of MA(1): \(Y_t = e_t + \theta_1 e_{t-1}\) \[\begin{align*} Var\Big(Y_t \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big) \ &= Var\Big( e_t + \theta_1 e_{t-1} \hspace{2mm} \Big| \hspace{2mm} Y_{t-1}, Y_{t-2}, \ldots, e_{t-1}, e_{t-2}, \ldots\Big)\\ &= Var( e_t ) \hspace{2mm} = \hspace{2mm} \sigma^2. \end{align*}\]
AR(1) \[\begin{align*} \mbox{ Uncond'l } \hspace{20mm} \mbox{ cond'l } \\ E(Y_t) &= 0, \hspace{28mm} E(Y_t|\omega_{t-1}) = \phi_1 Y_{t-1}, \\ Var(Y_t) &= (1+\phi_1^2)\sigma^2 \hspace{10mm} Var(Y_t|\omega_{t-1}) = \sigma^2 \end{align*}\]
MA(1) \[\begin{align*} \mbox{ Uncond'l } \hspace{20mm} \mbox{ cond'l } \\ E(Y_t) &= 0, \hspace{28mm} E(Y_t|\omega_{t-1}) = \theta_1 e_{t-1}, \\ Var(Y_t) &= (1+\theta_1^2)\sigma^2 \hspace{10mm} Var(Y_t|\omega_{t-1}) = \sigma^2 \end{align*}\]
For ARMA(p,q) model, conditional mean changes, but conditional variance is constant.

  x  <- arima.sim(n = 50, list(ar = c(0.8) )) 
  plot(x, type="o", main="AR(1)")
  abline(h=0)

  x  <- arima.sim(n = 50, list(ma = c(0.5) )) 
  plot(x, type="o", main="MA(1)")
  abline(h=0)

Heteroschedasticity

Don’t confuse the conditional heteroscedasticity with (unconditonal) heteroscedasticity:

  D <- read.csv("https://nmimoto.github.io/datasets/Gas.csv")
  D1 <- ts(D[,2], start=c(1956, 1), freq=12)
  plot(D1, type='o')

3. GARCH model

GARCH(1,1) model

\[\begin{align*} Y_t &= \sigma_t e_t \hspace{10mm} e_t \sim_{iid} N(0,1) \\\\ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \end{align*}\]

Conditional Mean and Var of GARCH

Conditional Mean: \(0\)
Conditional Variance : \(\sigma_t^2\)

__Ex: Daily SPY

Daily Price of SP500 ETF (SPY) from Jan 02 2000 to Dec 31 2014

  library(quantmod)
  source('https://nmimoto.github.io/R/TS-00.txt')

  getSymbols("SPY")    #- SP500 download from Yahoo!

## [1] "SPY"

  SPY <- Ad(SPY)['2010::']

  is.ts(SPY)      # not ts object

## [1] FALSE

  is.xts(SPY)     # its xts object

## [1] TRUE

  plot( SPY )

  plot( log(SPY) )

  plot( diff( log(SPY) ) )

  Randomness.tests( diff(log(SPY)) )

##   B-L test H0: the series is uncorrelated
##   M-L test H0: the square of the series is uncorrelated
##   J-B test H0: the series came from Normal distribution
##   SD         : Standard Deviation of the series

##      BL15 BL20 BL25 ML15 ML20 JB    SD
## [1,]    0    0    0    0    0  0 0.011

  # Fit GARCH model 
  library(fGarch)

  Y <- diff( log(SPY) )[-1]     # remove the first diff for NA
  
    Fit01 <-  garchFit(~ garch(1,1), data=Y, cond.dist="norm", include.mean = FALSE, trace = FALSE)
    Fit01

## 
## Title:
##  GARCH Modelling 
## 
## Call:
##  garchFit(formula = ~garch(1, 1), data = Y, cond.dist = "norm", 
##     include.mean = FALSE, trace = FALSE) 
## 
## Mean and Variance Equation:
##  data ~ garch(1, 1)
## <environment: 0x7ff778c68390>
##  [data = Y]
## 
## Conditional Distribution:
##  norm 
## 
## Coefficient(s):
##      omega      alpha1       beta1  
## 3.7778e-06  1.7597e-01  7.8801e-01  
## 
## Std. Errors:
##  based on Hessian 
## 
## Error Analysis:
##         Estimate  Std. Error  t value Pr(>|t|)    
## omega  3.778e-06   5.531e-07    6.830 8.49e-12 ***
## alpha1 1.760e-01   1.767e-02    9.959  < 2e-16 ***
## beta1  7.880e-01   1.815e-02   43.428  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Log Likelihood:
##  8734.112    normalized:  3.381383 
## 
## Description:
##  Thu Apr  9 15:38:43 2020 by user:

    ##  Fit01@fit$par                  # estimated parameters
    ##  Fit01@residuals                # this is not GARCH residuals! This is same as Y.
    ##  Fit01@sigma.t                  # estimated sig_t
    ##  Fit01@residuals/Fit1@sigma.t   # this is the (standardized) GARCH residuals
  
    Fit01@fit$ics                   # AIC and BIC are here

##       AIC       BIC       SIC      HQIC 
## -6.760443 -6.753641 -6.760445 -6.757977

    plot(Y)

    plot(Fit01@sigma.t, type="l")   # estimated Sigma_t

    sigma.upper = xts(1.96*Fit01@sigma.t, order.by=index(Y))
    sigma.lower = xts(-1.96*Fit01@sigma.t, order.by=index(Y))
    
    plot( cbind(Y, sigma.upper, sigma.lower),
        col=c("black", "red", "red"), lwd=c(2,1,1) )

    plot(  as.numeric(          Y["2018::"]), type="h", lwd=2)
    lines( as.numeric(sigma.upper["2018::"]), col="red")
    lines( as.numeric(sigma.lower["2018::"]), col="red")

GARCH(1,1) model

\[\begin{align*} Y_t &= \sigma_t e_t \hspace{10mm} e_t \sim_{iid} N(0,1) \\\\ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2 + \beta \sigma^2_{t-1} \end{align*}\]
standardized residuals

\[\begin{align*} \hat \sigma_t^2 &= \hat \omega + \hat \alpha Y_{t-1}^2 + \hat \beta \sigma^2_{t-1} \\ \\ \hat e_t &= \frac{Yt}{\hat \sigma} \end{align*}\]

    res1  <- Y/Fit01@sigma.t        #- this is GARCH residuals
    Randomness.tests(res1)

##   B-L test H0: the series is uncorrelated
##   M-L test H0: the square of the series is uncorrelated
##   J-B test H0: the series came from Normal distribution
##   SD         : Standard Deviation of the series

##       BL15  BL20  BL25  ML15 ML20 JB    SD
## [1,] 0.403 0.272 0.063 0.463 0.74  0 0.998

TS Class Web Page – R resource page

9. GARCH model