TD1 SER2

Consider the following AR(2) random process \[ X_t = 40 + 0.4 X_{t-1} - 0.2 X_{t-2} + \epsilon_t \quad \text{where } \epsilon_t \sim \mathcal{WN}(0, \sigma^2 = 12.8). \]

1. Verify that the process is stationary.
2. Compute the expectation of \(X_t\).
3. Give the Yule-Walker equations satisfied by the autocovariances of the process and compute the variance, as well as the first 5 values of the autocorrelations.
4. Show that the partial autocorrelation function satisfies \( r(h) = 0 \) if \( h \geq 3 \).
5. Compute the first three partial autocorrelations.

Let \(Y_{t}\) be an \(AR(2)\) process: \[ Y_{t} = -\frac{1}{2}Y_{t-1} + \frac{7}{50}Y_{t-2} + \varepsilon_{t} \] where \(\varepsilon_{t}\) is a white noise with \((\sigma^{2} = 1)\).

1. Is the process \(Y_{t}\) stationary and causal?
2. Give the transfer function \(\psi(B)\) expressing \(Y_{t}\) as \(Y_{t} = \psi(B)\varepsilon_{t}\).
3. Give a moving average representation \( Y_{t} = \sum_{i=0}^{\infty} \psi_{i} \varepsilon_{t-i} \), specifying the coefficients \(\psi_{i}\).
4. Compute the correlation \(\rho_{1}\) for this stationary process.
5. Give a recurrence for \(\rho_{k}\), \(k \geq 2\), and find \(\rho_{2}\), then solve this recurrence for \(\rho_{k}\).

We have (for all \(n\)) \(\varepsilon_{n} \sim WB(0,\sigma^{2})\) and \[ X_{n} = (1/6)X_{n-1} + (1/6)X_{n-2} + \varepsilon_{n}. \]

1. Is this process stationary? Causal? Invertible? Justify!
2. Deduce that \( X_{n} = \sum_{i=0}^{\infty} a_{i} \varepsilon_{n-i}\) and hence \(\operatorname{cov}(\varepsilon_{n}, X_{n-1}) = 0\) and \(\operatorname{cov}(\varepsilon_{n}, X_{n-2}) = 0\).
3. Demonstrate the Yule-Walker equations for the autocovariance in the general case of an AR(\(p\)) process.
4. Use the previous question to determine \(\gamma(0)\), \(\gamma(1)\) and give the general formula for \(\gamma(h)\) as a function of \(h\) and \(\sigma^{2}\).
5. Give the \(MA(\infty)\) representation.

Let \(\varepsilon\) be a weak white noise and let \(X\) be a second-order stationary process satisfying: \[ X_{t} + 0.4X_{t-1} - 0.12X_{t-2} = \varepsilon_{t}. \]

1. What type of ARMA model is this?
2. Show that \(\varepsilon\) is the innovation white noise associated with \(X\).
3. Deduce that \(\operatorname{cov}(\varepsilon_{t}, X_{s}) = 0\) for all \(s < t\), and then that the autocorrelation function of \(X\) satisfies the recurrence relation: \(\rho_{X}(t) = -0.4\rho_{X}(t-1) + 0.12\rho_{X}(t-2), \quad \forall t \geq 2.\)
4. Show that \(\rho_{X}(t) = \frac{2}{11}(0.2)^{t} + \frac{9}{11}(-0.6)^{t}, \quad \forall t \geq 2.\)
5. Determine the partial autocorrelation function of \(X\).
6. Give the forecast for \(X_{T+h}\) as a function of \(h\), \(X_{t}\), and \(X_{t-1}\).

On monthly data, consider the representation \(x_{t} = (1 - \theta_{1}L^{2})(1 - \theta_{2}L^{12})u_{t}\), where \(u_{t}\) is a white noise process.

1. Give the values of the first 15 autocorrelation coefficients of the series \(x_{t}\).
2. Let \(\theta_{1} = 0.8\) and \(\theta_{2} = -1.1\). Is the process stationary?
3. Now set \(\theta_{1} = 0.8\) and \(\theta_{2} = -0.8\). Give the first 5 values of the autoregressive representation of the process.
4. Give the general formula for the forecast of \(x_{t+h}\) and a 95% confidence interval.

Consider the process \[ X_{t} - \frac{7}{6}X_{t-1} + \frac{1}{3}X_{t-2} = \varepsilon_{t} - \frac{1}{4}\varepsilon_{t-1} - \frac{1}{8}\varepsilon_{t-2}. \]

1. What type of model is this?
2. Is this process stationary? Causal? Invertible?
3. Propose another representation of this model of the form \((1 - \lambda_{1}L)(1 - \lambda_{2}L)X_{t} = (1 - \theta_{1}L)(1 - \theta_{2}L)\varepsilon_{t}\), where \(\lambda_{1}\), \(\lambda_{2}\), \(\theta_{1}\), \(\theta_{2}\) are non-zero real numbers, not necessarily distinct. Conclude!
4. Give the \(MA(\infty)\) and \(AR(\infty)\) representations of this process.
5. Give the autocovariance function.

Soit \((X_{t}, t\in \mathbb{Z})\) un processus \(ARMA(1,1)\) satisfaisant l'équation \[ X_{t} -\varphi X_{t-1} = u_{t} + \theta u_{t-1};\quad u_{t}\sim BB(0,\sigma^{2}) \] où \(|\varphi|< 1\) et \(|\theta| < 1\).

1. \(X_{t}\) est-il stationnaire? Pourquoi?
2. \(X_{t}\) est-il inversible? Pourquoi?
3. Calculer \(E (X_{t})\).
4. Déterminez les coefficients de \(u_{t}, u_{t-1}, u_{t-2}, u_{t-3}\) et \(u_{t-4}\) de la représentation MA.
5. Déterminez la fonction d'autocorrélation totale de \(X_{t}\).
6. Si \(\varphi= 0.5, \theta= 0.5, X_{10} = 1.0, u_{10}= 0.5\) et \(X_{t}= u_{t}= 0 \) pour \(t\leq 9\), donnez des prévisions optimales (au sens de l'erreur quadratique moyenne) pour \(X_{11}\) et \(X_{12}\) et donner la formule générale de \(\widehat{X}_{t+h}\).

Consider the following representation: \[ (1 - 0.3L) x_{t} = 2.0 + u_{t} - 0.7 u_{t-1} + 0.12 u_{t-2} \] where \(u_{t}\) is a Gaussian white noise with variance \(\sigma^{2}_{u} = 2\).

At time \(T\), we know the last 4 realizations of \(x\) and \(u\). These are given in the table below.

1. Is this process stationary? Invertible?
2. Specify the first 5 values of the autocorrelation function of \(x\).
3. Give the optimal forecasts for \(x\) at dates \(T+1\) and \(T+2\) with a 95% confidence interval.
4. Give the general formula for \(\widehat{X}_{t+h}\) where \(h\) is the forecast horizon, as a function of \(X_{t}, X_{t-1}, \ldots\)
5. What is the probability that \(x\) takes a negative value?

Consider the process \(\phi(L) y_{t} = \Theta(L)u_{t}\), where \(u_{t}\) is a white noise process with variance \(\sigma^{2}_{u}\), \(\phi(L) = (1 - \varphi L^{2})\), \(\Theta(L) = (1 - \theta L)\), with \(\varphi\) and \(\theta\) real numbers and \(L\) the usual lag operator.

1. State the conditions for stationarity and invertibility of the process \(y\).
2. Assuming the previous conditions are satisfied, give the expressions for the first three autocovariances of \(y\): \(\gamma_{0}\), \(\gamma_{1}\), \(\gamma_{2}\) as functions of the process parameters \(\varphi\), \(\theta\) and \(\sigma^{2}_{u}\).
3. Give the general formula for \(\widehat{y}_{T+h}\).

Let \(X\) be a stationary \(AR(1)\) defined by: \[ \forall t\in\mathbb{Z},\quad X_{t}=\phi X_{t-1}+Z_{t}, \] where \(Z\) is a centered white noise with variance \(\sigma^{2}\) and \(|\phi|<1\). Define: \(\forall t\in \mathbb{Z},\quad Y_{t}=X_{t}-X_{t-1}.\)

1. Show that \(Y\) is a stationary centered process, and that it is an \(ARMA(p,q)\). Specify the values of \(p\) and \(q\) and give the recurrence equation satisfied by \(Y\).
2. What is the variance of \(Y_{t}\) (for any \(t\))?
3. Give the autocovariance function of \(Y\).
4. Give the forecast of \(Y_{t+h}\) given the observations \(X_{t}\), \(X_{t-1}\), \(X_{t-2}\), \ldots and \(Z_{t}\), \(Z_{t-1}\), \ldots
5. How should we proceed if we only have the data: \(X_{t}, X_{t-1}, X_{t-2}, \ldots\)?

GDP per capita, \(Y_{t}\), is modeled as: \[ Y_{t} = 10 + 0.02t + X_{t} \quad \text{with} \quad (1 - 0.9L)X_{t} = \varepsilon_{t} \] and \(\varepsilon_{t}\) is a white noise with standard deviation \(\sigma_{\varepsilon} = 0.06\).

1. What type of model is this?
2. At time \(t = 80\), we observe \(Y_{80} = 11.2\). Do you conclude that the economy at this date is rather in a phase of cyclical slowdown or rather in a period of boom? What forecast do you make at date \(t = 80\) for \(Y_{81}\)? Give a 95% confidence interval for your forecast.
3. Show that the conditional variance of a stationary process approaches its unconditional variance.
4. Still at date \(t = 80\), you are now asked to make a forecast for \(Y_{160}\). What is your answer? And what is the 95% confidence interval?

Consider the model: \[ (I - \varphi L)(I - L)Y_{t} = (I + \theta L)\varepsilon_{t}, \] with \(|\varphi| < 1\) and \(|\theta| < 1\).

1. Show that \(\widehat{Y}_{T+1} = (1 + \varphi)Y_{T} - \varphi Y_{T-1} + \theta \varepsilon_{T}\) and \(\widehat{Y}_{T+k} = (1 + \varphi) \widehat{Y}_{T+(k-1)} - \varphi \widehat{Y}_{T+(k-2)}, \quad \text{for } k \geq 2.\)
2. Show that for \(k \geq 0\), \(\widehat{Y}_{T+k} = A + B \varphi^{k}\) and find expressions for \(A\) and \(B\) in terms of \(Y_{T}\), \(Y_{T-1}\), \(\varepsilon_{T}\), \(\varphi\), and \(\theta\), using the initial conditions \(\widehat{Y}_{T}(0) = Y_{T}\) and \(\widehat{Y}_{T+1}\).
3. Show that \(\widehat{Y}_{T+k} = Y_{T} + \varphi \left( \frac{1 - \varphi^{k}}{1 - \varphi} \right)(Y_{T} - Y_{T-1}) + \theta \left( \frac{1 - \varphi^{k}}{1 - \varphi} \right) \varepsilon_{T}, \quad k \geq 0.\)
4. Find the limit of \(\widehat{Y}_{T+k}\) as \(k \to \infty\).

The amount of Gross Domestic Product (GDP) was modeled using a linear regression and we obtained: \[ \widehat{PIB}_{t} = 33 + 1.90 t + \widehat{u}_{t}, \quad \text{with } t = 1, 2, 3, \ldots \] The variance of the errors, assumed Gaussian, is estimated as \(\sigma_{u}^{2} = 120\). On these errors, an MA(3) was estimated: \[ (1 + 1.26L + 1.11L^{2} + 0.61L^{3})\widehat{u}_{t} = \varepsilon_{t}. \] Finally, at period \(t = 240\), we observe \(PIB_{240} = 506\). Using this information:

1. Would you say that we are in a recession period at \(t = 240\) and why?
2. What type of model is this?
3. Identify the nature of the trend(s) present in \(\widehat{PIB}_{t}\) (stochastic, deterministic)?
4. At \(t = 240\), you are also asked to construct a forecast for \(t = 243\) and provide a 95% confidence interval for this forecast. Briefly explain your approach.

Consider the process modeled by \(Y_{t} = \beta t + S_{t} + U_{t}\) where \(\beta \in \mathbb{R}\), \(S_{t}\) is a periodic function with period 4, and \(U = (U_{t})_{t\in\mathbb{Z}}\) is a stationary process.

1. Is the process \((Y_{t})_{t\in \mathbb{Z}}\) stationary?
2. Show that \(Z = (1 - B^{4})Y\) is stationary and compute its autocovariance as a function of that of \(U\).
3. Given the previous question, what type of model is this?
4. Conclude the filter that allows removing seasonality of order \(p\).
5. In general, if \(X_t\) is a time series whose trend term is defined by a polynomial of degree \(n\), show that the differenced series \((1 - L)^{n} X_t\) is of the form \((1 - L)^{n} X_t = a + \varepsilon_t\), where \(\varepsilon_t\) is a sequence of random variables with zero mean and constant variance with respect to \(t\).