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exercises/fundamental_concepts_univariate_time_series.tex
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exercises/fundamental_concepts_univariate_time_series_solution.tex
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -1,62 +1,64 @@ | ||
| \begin{enumerate} | ||
| \item A white noise has mean zero, a constant variance and all other second-order moments (i.e. autocovariances/autocorrelations) are zero: | ||
| \item A white noise has mean zero, a constant variance and all other second-order moments (i.e.\ autocovariances/autocorrelations) are zero: | ||
| \begin{align*} | ||
| E[\varepsilon_t]&=0\\ | ||
| Var[\varepsilon_t]&=E[\varepsilon_t^2] - E[\varepsilon_t]E[\varepsilon_t] = \sigma_\varepsilon^2\\ | ||
| Cov(\varepsilon_{t},\varepsilon_s) &= E[\varepsilon_t \varepsilon_s] - E[\varepsilon_t]E[\varepsilon_s] = 0 \text{ for $s \neq t$} | ||
| Cov(\varepsilon_{t},\varepsilon_s) &= E[\varepsilon_t \varepsilon_s] - E[\varepsilon_t]E[\varepsilon_s] = 0~\text{for}~s \neq t | ||
| \end{align*} | ||
| \item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/whiteNoisePlots.m} | ||
| Every simulation is different, model can thus generate an infinite set of realizations over the period $t=1,...,200$. | ||
| Every simulation is different, model can thus generate an infinite set of realizations over the period \(t=1,\ldots ,200\). | ||
| The processes do differ in their persistence. | ||
| (i) is the white noise process, which is not persistent. | ||
| (ii) is a 5-point-moving-average, which is a linear combination of white noise processes. | ||
| It is smoother and more persistent and very different from just noise. | ||
| Linear combinations of white noise processes build the basis of many models in time series analysis. | ||
| \item A process is said to be \textbf{$N$-order weakly stationary} if all its joint moments up to order $N$ exist and are time invariant. | ||
| We are particularly interested in $N=2$, i.e. \textbf{covariance stationarity}: | ||
| \item A process is said to be \textbf{\(N\)-order weakly stationary} if all its joint moments up to order \(N\) exist and are time invariant. | ||
| We are particularly interested in \(N=2\), i.e.\ \textbf{covariance stationarity}: | ||
| \begin{align*} | ||
| E[Y_t]&=\mu \text{ (constant for all t)} | ||
| E[Y_t]&=\mu~\text{(constant for all t)} | ||
| \\ | ||
| Var[Y_t]&=E[(Y_t - \mu)(Y_t-\mu)]=\gamma_0 \text{ (constant for all t)} | ||
| Var[Y_t]&=E[(Y_t - \mu)(Y_t-\mu)]=\gamma_0~\text{(constant for all t)} | ||
| \\ | ||
| Cov[Y_{t_1},Y_{t_1-k}] &= E[(Y_{t_1}-\mu)(Y_{t_1-k}-\mu)] = Cov[Y_{t_2},Y_{t_2-k}] = \gamma_k \text{ (only dependent on $k$)} | ||
| Cov[Y_{t_1},Y_{t_1-k}] &= E[(Y_{t_1}-\mu)(Y_{t_1-k}-\mu)] = Cov[Y_{t_2},Y_{t_2-k}] = \gamma_k~\text{(only dependent on \(k\))} | ||
| \end{align*} | ||
| That is the first two moments are not dependent on $t$. | ||
| Particularly, the autocovariance is only dependent on the time difference $k$, but not on the actual point in time $t$. | ||
| That is the first two moments are not dependent on \(t\). | ||
| Particularly, the autocovariance is only dependent on the time difference \(k\), but not on the actual point in time \(t\). | ||
| \\ | ||
| \textbf{Strict stationarity}: for all $k$ and $h$: $$f(Y_t,Y_{t-1},...,Y_{t-k})=f(Y_{t-h},Y_{t-h-1},...,Y_{t-h-k})$$ | ||
| That is, not only the first two moments but the whole distribution is not dependent on the point in time $t$, | ||
| but on the time difference $k$. | ||
| \textbf{Strict stationarity}: for all \(k\) and \(h\): | ||
| \[f(Y_t,Y_{t-1},\ldots ,Y_{t-k})=f(Y_{t-h},Y_{t-h-1},\ldots ,Y_{t-h-k})\] | ||
| That is, not only the first two moments but the whole distribution is not dependent on the point in time \(t\), | ||
| but on the time difference \(k\). | ||
| \item Autocovariance function for a covariance-stationary process: | ||
| $$\gamma_k = E[(Y_t - \mu)(Y_{t-k}-\mu)]$$ | ||
| where $\gamma_0$ is the variance. Autocorrelation function: $$\rho_k = \gamma_k/\gamma_0$$ | ||
| \\ | ||
| \[\gamma_k = E[(Y_t - \mu)(Y_{t-k}-\mu)]\] | ||
| where \(\gamma_0\) is the variance. Autocorrelation function: \[\rho_k = \gamma_k/\gamma_0\] | ||
| \\ | ||
| We can estimate this by using: | ||
| \begin{align*} | ||
| \hat{\gamma}_k = c_k &= \frac{1}{T} \sum_{t=k+1}^T(y_t -\bar{y})(y_{t-k}-\bar{y})\\ | ||
| \hat{\rho}_k = r_k & = c_k/c_0 | ||
| \end{align*} | ||
| \hat{\gamma}_k = c_k &= \frac{1}{T} \sum_{t=k+1}^T(y_t -\bar{y})(y_{t-k}-\bar{y}) | ||
| \\ | ||
| \hat{\rho}_k = r_k & = c_k/c_0 | ||
| \end{align*} | ||
| Note: In most applications we don't correct the degrees of freedom for numerical reasons | ||
| (e.g. to avoid singularity of autocovariance matrices in the multivariate case), | ||
| i.e. the sums are not divided by $T-k-1$ but simply by $T$. | ||
| For $T>100$ this does not really matter as the expressions are very close to each other. | ||
| (e.g.\ to avoid singularity of autocovariance matrices in the multivariate case), | ||
| i.e.\ the sums are not divided by \(T-k-1\) but simply by \(T\). | ||
| For \(T>100\) this does not really matter as the expressions are very close to each other. | ||
|
|
||
| \item \lstinputlisting[style=Matlab-editor,basicstyle=\mlttfamily,title=\lstname]{progs/matlab/plotsAR1.m} | ||
| Remarks: If $|\phi|<1$ the series returns to the mean, i.e. it is stable and stationary. | ||
| If $|\phi>1|$ then it explodes, i.e. it is unstable and not stationary. | ||
| $\phi=1$ is a so-called random walk, | ||
| Remarks: If \(|\phi|<1\) the series returns to the mean, i.e.\ it is stable and stationary. | ||
| If \(|\phi>1|\) then it explodes, i.e.\ it is unstable and not stationary. | ||
| \(\phi=1\) is a so-called random walk, | ||
| it is the key model when working with non-stationary models. | ||
| Note that the random walk incorporates many different shapes, in macroeconomic forecasts we often want to \textbf{beat} the random walk model. | ||
| \item It is a special LINEAR operator, similar to the expectation operator, and very useful when working with time series. | ||
| The operator transforms one time series into another by shifting the observation from period $t$ to period $t-1$: | ||
| $Ly_t = y_{t-1}$ or $L^{-1} y_t =y_{t+1}$. | ||
| More general: $L^k y_t = L^{k-1} L y_t = L^{k-1} y_{t-1} = ... = y_{t-k}$. | ||
| The operator transforms one time series into another by shifting the observation from period \(t\) to period \(t-1\): | ||
| \(Ly_t = y_{t-1}\) or \(L^{-1} y_t =y_{t+1}\). | ||
| More general: \(L^k y_t = L^{k-1} L y_t = L^{k-1} y_{t-1} = \cdots = y_{t-k}\). | ||
| Convenient use: | ||
| $$(1-L)y_t = y_t - y_{t-1}= \Delta y_t$$ | ||
| \[(1-L)y_t = y_t - y_{t-1}= \Delta y_t\] | ||
| We can also work with lag-polynomials: | ||
| $$ \phi(L) = (1-\phi_1 L-\phi_2 L^2 -... - \phi_p L^p)$$ | ||
| \[ \phi(L) = (1-\phi_1 L-\phi_2 L^2 -\cdots - \phi_p L^p)\] | ||
| where we call p the lag order. So: | ||
| $$ \phi(L) y_t = (1-\phi_1 L-\phi_2 L^2 -... - \phi_p L^p)y_t = y_t - \phi_1 y_{t-1} -\phi_2 y_{t-2} - ... - \phi_p y_{t-p}$$ | ||
| To check whether an AR(p) model is covariance stationarity, we need to check whether the roots of the lag-polynomial lie outside the unit circle. | ||
| That is, we treat $L$ as a complex number $z\in \mathbb{C}$ and compute the roots of $(1-\phi_1 z-\phi_2 z^2 -... - \phi_p z^p)=0$ (using a computer in most cases). | ||
| \[ \phi(L) y_t = (1-\phi_1 L-\phi_2 L^2 -\cdots - \phi_p L^p)y_t = y_t - \phi_1 y_{t-1} -\phi_2 y_{t-2} - \cdots - \phi_p y_{t-p}\] | ||
| To check whether an {AR{(p)}} model is covariance stationarity, we need to check whether the roots of the lag-polynomial lie outside the unit circle. | ||
| That is, we treat \(L\) as a complex number \(z\in \mathbb{C}\) and compute the roots of \((1-\phi_1 z-\phi_2 z^2 -\cdots - \phi_p z^p)=0\) (using a computer in most cases). | ||
| \end{enumerate} |
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