Time Series statistical models

Time series statistical models

5-6 Intro

• Adjecent data points in time series have strong correlation
• At discrete time points (t=0, ±1, ±2…) we make observstion ($X0$, $X{±1}$, $X_{±2}$, …) about something
  • $X$ is the random variable itself
  • $x0$, $x{±1}$, … is also time series
  • t=0 point is decided by our own

7 White noise series

• $W0, W{±1}, W_{±2}$, …
• $Wt ∼ WN[0, \sigmaW^2]$
• WN → N: Gaussian white noise
• All W_t are uncorrelated and independent

Measures of dependence

Mean function

• Mean is evaluated at a given TIME POINT
  • On that time point t, mean is calculated over $X_t$ by using integral
  • eg. The average number of patients on Tuesday
• Random walk model
  • Expression:
  • $X_0 = 0$                     for $t=0$
  • $Xt = X{t-1}+W_t$ for $t = 1, 2, …$
  • $Wt ∼ N[0, \sigmaW^2]$
  • $\mu = 0$

Autocorrelation function

• $s, t$: two time points
• Autocovariance
  • $\gammaX(s,t) = Cov(Xs, Xt) = E[(Xs-\muX(s))(Xt-\mu_X(t))]$
  • When s=t, $\gammaX(t,t) = Var(Xt)$
• 12 Autocorrelation
  • $\Large\rhoX(s,t) = \frac{\gammaX(s,t)}{\sqrt{\gammaX(s,s)\gammaX(t,t)}}$
  • When s=t, $\rho_X(t,t) = 1$
  • if $\rho$ always approach 1 for some points $s < t$, then $Xt$ can be predicted by $Xs$ by a linear model $Xt = \beta0+\beta1Xs+Wt$ , $Wt～N(0,\sigma_W^2)$
• 13 Crosscorrelation
  • $\Large\rho{X,Y}(s,t) = \frac{\gamma{X,Y}(s,t)}{\sqrt{\gammaX(s,s)\gammaY(t,t)}}$
  • if $\rho$ always approach 1 for some points $s < t$, then $Yt$ can be predicted by $Xs$ by a linear model $Yt = \beta0+\beta1Xs+Wt , Wt～N(0,\sigma_W^2)$

Stationary time series

Criteria

• $\mu_X(t)$ is constant
• $\gamma_X(s,t)$ is solely determined by $h = |s-t|$
  • when s=t, the result is var, and it is constant, not dependent on t
  • The key issue is to decouple the role of $t$

Characteristics

• Auto…shits at lag h, derived from 11-13 part:
  • $\gammaX(h) = Cov(X{t+h}, X_t)$
  • $Var(Xt) = \gammaX(0)$
  • $\Large\rho_X(h) = \frac{\gamma(h)}{\gamma(0)}$
• Gaussian white noise is stationary because:
  • $\mu=0$, by definition
  • $\gamma(s,t)$:
  • s=t:
    • $\gamma(s,t) = Var(Wt) = \sigmaW^2$
  • s≠t:
    • $\gamma(s,t) = Cov(Xs, Xt) = 0$ because they are uncorrelated
• Random walk model is not stationary because:
  • $\mu=0$, by definition
  • $\gamma(t,t) = Cov(Xt,Xt) = Var(Xt) = Var(W1+…+Wt) = Var(W1) + … + Var(Wt) + Cov(W1,W2) + … + Cov(Wt-1,Wt) = t*\sigma^2 + 0$
  • gamma thing is dependent of time t so its not stationary
• Homoskedasticity: variance is constant over time
• Heteroskedasticity: variance changes over time
• 23 Statistical hypothesis test
• 24 Jointly stationary time series
• 25 example:
  • $|\rho|$ is largest at $h=-6$, $Yt = \beta0 + \beta1*X{t-6} + W_t$
  • Can predict Yt based on Xt from 6 months ago

Ass 1

• $\mu_X(t)$
  • $\muX(t) = E(Xt) = E(W0 + W1 + … + Wt) = 0$
  • Satisfies this criteria
• $\gamma(s,t)$ Let s=t. Since $W_t$Then: $\gamma(s,t)$ $= Cov(Xt,Xt)$ $= Var(X_t)$ $= Var(W0+…+Wt) $ $= Var(W0) + … + Var(Wt) + Cov(W0,W1) + Cov(W0,W2) + … + Cov(W{t-1},Wt)$

       $= (t+1)\sigma_W^2 + 0$

        $\gamma(s,t)$ is dependent on $t$ when $s=t$

In conclusion, that time series model is NOT stationary.