\newcommand{\backshift}{\operatorname{B}} \newcommand{\var}{\operatorname{Var}} \newcommand{\expect}{\operatorname{E}} \newcommand{\corr}{\operatorname{Corr}} \newcommand{\cov}{\operatorname{Cov}} \newcommand{\ssr}{\operatorname{SSR}} \newcommand{\se}{\operatorname{SE}}

\newcommand{\mat}[1]{\bm{#1}} \newcommand{\eval}[2]{\left. #1 \right\vert_{#2}}

Problem 1.1

Suppose that simple exponential smoothing is being used to forecast the process $y_t = \mu + e_t$, where where ${e_t}$ are white noise with mean $0$ and variance $\sigma^2$. At the start of period $t^{∗}$, the mean of the process experiences a transient; that is, it shifts to a new level $\mu + \delta$, but reverts to its original level $\mu$ at the start of the next period $t^{∗} + 1$. The mean remains at this level for subsequent time periods.

Part (a)

\fbox{\begin{minipage}{.8\textwidth} Find the expected value of the simple exponential smoother

\end{minipage}}

We have a time series

except at $y_{t^*}$ which is distributed

where the error terms are zero mean white noise with variance $\sigma^2$.

The expectation of the smoothed time series $\tilde{y}T$ is given by \begin{align*} \expect(\tilde{y}T) &= (1-\theta)\sum{t=0}^{\infty} \theta^t \expect(y{T-t})\ &= (1-\theta)\left( \sum_{t=0}^{T-t^+1} \theta^t \mu + \theta^{T-t^}(\mu + \delta) + \sum_{t=T-t^-1}^{\infty} \theta^t \mu \right)\ &= (1-\theta)\left( \sum_{t=0}^{\infty} \theta^t \mu + \theta^{T-t^} \delta \right)\ &= \mu + (1-\theta) \theta^{T-t^} \delta. \end{align}

Part (b)

\fbox{\begin{minipage}{.8\textwidth} For $\theta = 0.5$, determine the number of periods that it will take following the impulse for the expected value of $\tilde{y}_T$ to return to within $0.1\delta$ of the original level $\mu$. \end{minipage}} We wish to find $\tilde{y}_k$ such that it is expected to be within $\frac{1}{10} \delta$ of $\mu$,

Plugging in the definition of the expectation and simplifying,

Since pulling all positive numbers (or symbols that stand for positive numbers) out of the absolute value function does not change the expression, we may rewrite the above as

Dividing by $\vert \delta \vert$ on both sides,

which may be rewritten as

Taking the logarithm of both sides \begin{align*} k \log \theta &\leq \log \left(\frac{\theta^{t^}}{10(1-\theta)}\right)\ &\leq t^ \log \theta - \log 10 - \log(1-\theta). \end{align*}

Finally, we isolate $k$ by dividing by $\log \theta$ on both sides. However, note that $\log \theta$ is negative, and so we must flip the inequality,

Letting $\theta = 0.5$,

which simplifies to

We wish to take the \emph{smallest} $k$ that is an integer that satisfies the equation. Thus, $k = t^* + 3$. Or, in other words, $3$ periods after $t^*$, $\tilde{y}_T$ has an expectation that is within the specified distance of $\mu$.

Problem 1.2

Let ${Y_t}$ be an AR(1) process with $|φ| < 1$. That is $Y_t = φY_{t−1} + e_t$, where ${e_t}$ are white noise with mean $0$ and variance $\sigma^2$. Also note $e_t$’s are independent of $Y_{t−1}, Y_{t−2},\ldots$.

Part (a)

\fbox{\begin{minipage}{.8\textwidth} Find the autocorrelation function for $W_t = Y_t - Y_{t-1}$ in terms of $φ$ and $\sigma^2$. \end{minipage}}

Observe that

and thus

The autocovariance function for $W_t$, denoted by $γ_{{W_t}}$, is defined as

Assuming $k \neq 0$ (we solve directly for variance in that case) and replacing $W_t$ and $W_{t-k}$ with their respective definitions yields \begin{align*} γ_{{W_t}}(k) &= \cov((φ - 1) Y_{t-1} + e_t,(φ - 1) Y_{t-k-1} + e_{t-k})\ &= \cov((φ - 1) Y_{t-1},(φ - 1) Y_{t-k-1})\ &= (φ - 1)^2 \cov(Y_{t-1},Y_{t-k-1}). \end{align*}

Observe that $\cov(Y_{t-1},Y_{t-k-1}) = γ_{{Y_t}}(k)$. Since ${Y_t}$ is $\operatorname{AR}(1)$,

Thus,

The variance of ${W_t}$ is given by \begin{align*} \cov(W_t,W_t) &= \cov((φ - 1) Y_{t-1} + e_t, (φ - 1) Y_{t-1} + e_t)\ &= (φ - 1)^2 \cov(Y_{t-1},Y_{t-1}) + \cov(e_t,e_t)\ &= (φ - 1)^2 \frac{\sigma^2}{1-φ^2} + \sigma^2\ &= \sigma^2\left(1 + \frac{(φ - 1)^2}{1-φ^2}\right). \end{align*}

Thus, the autocorrelation function is given by

which simplifies to

Part (b)

In part (a), we found that

Problem 1.3

Suppose $Y_t = X_t + e_t$, where ${e_t}$ are normal white noise with mean $0$ and variance $\sigma_e^2$. The ${X_t}$ process is a stationary AR(1) defined by $X_t = φX_{t−1} + Z_t$, where ${Z_t}$ is a zero mean normal white noise process with variance $\sigma_Z^2$. As usual, in the AR(1) process, assume that $Z_t$ is independent of $X_{t−1}, X_{t−2}, \ldots$. Assume additionally that $\expect(e_t Z_s) = 0$ for all $t$ and $s$.

Part (a)

\fbox{Show that ${Y_t}$ is stationary and find its autocovariance function, $γ_k$.}

To be stationary, ${Y_t}$ must have a constant mean and a a autocovariance that is strictly a function of the lag.

The mean is given by

Since $X_t$ is AR(1) with mean $\delta / (1-φ) = 0$, we see that $\expect(Y_t) = 0$, i.e., is a constant zero.

The variance is given by

Since $X_t$ is AR(1), its variance is $\sigma_Z^2/(1-φ^2)$, thus

The autocovariance of ${Y_t}$ is given by

Since ${Y_t}$ has a constant expectation of zero, this simplies to

Observe that $Y_t = φ X_{t-1} + Z_t + e_t$ and

The expectation of $Y_t Y_{t-k}$ is given by \begin{align*} \expect(Y_t Y_{t-k}) &= φ \expect(Y_{t-k} X_{t-1}) + \expect(Y_{t-k} Z_t) + \expect(Y_{t-k} e_t)\ &= φ \expect(Y_{t-k} X_{t-1}) + \expect(Y_{t-k}) \expect(Z_t) + \expect(Y_{t-k}) \expect(e_t)\ &= φ \expect(Y_{t-k} X_{t-1})\ &= φ \expect((X_{t-k} + e_t) X_{t-1})\ &= φ \expect(X_{t-1} X_{t-k} + e_t X_{t-1})\ &= φ \left(\expect(X_{t-1} X_{t-k}) + \expect(e_t X_{t-1})\right)\ &= φ \expect(X_{t-1} X_{t-k}). \end{align*}

Since ${X_t}$ is AR(1), observe that the autocovariance function for ${X_t}$ is $γ_{{X_t}}(k) = φ \expect(X_{t-1} X_{t-k})$, which has a closed-form solution \begin{equation} γ_{{X_t}}(k) = \begin{cases} \frac{\sigma_Z^2}{1-φ^2} & k = 0\ φ γ_{{X_t}}(k-1) & k > 0. \end{cases} \end{equation}

Thus, the autocovariance function for ${Y_t}$ is given by \begin{equation} γ_k = \begin{cases} \frac{\sigma_Z^2}{1-φ^2} + \sigma_e^2 & k = 0\ γ_{{X_t}}(k) & k > 0. \end{cases} \end{equation}

Since its autocovariance function is strictly a function of lag and its mean is a constant zero, ${Y_t}$ is stationary. Note that it is not just weakly stationary, but strongly stationary given the normally distributed random errors.

Part (b)

\fbox{\begin{minipage}{.8\textwidth} Show that the process ${U_t}$, where $U_t = Y_t − φY_{t−1} = (1 − φB)Y_t$, has nonzero correlation only at lag 1 (excluding lag 0, of course!). \end{minipage}}

The autocovariance is given by \begin{align*} γ_{{U_t}}(k) &= \cov(U_t,U_{t-k})\ &= \cov(Y_t - φ Y_{t-1},Y_{t-k} - φ Y_{t-k-1}). \end{align*}

Observe that $Y_t - φ Y_{t-1} = X_t + e_t - φ(X_{t-1} + e_{t-1})$. Since $Z_t = X_t - φ X_{t-1}$, we see that

and

Thus,

If $k > 1$, then $γ_{{X_t}}(k) = \cov(e_t + Z_t - φ e_{t-1}, e_{t-k} + Z_{t-k} - φ e_{t-k-1}) = 0$ since they have no terms in common. If $k=1$, then \begin{align*} γ_{{X_t}}(1) &= \cov(e_t + Z_t - φ e_{t-1}, e_{t-1} + Z_{t-1} - φ e_{t-2})\ &= \cov(-φ e_{t-1}, e_{t-1})\ &= -φ \var(e_{t-1})\ &= -φ \sigma_e^2, \end{align*} which is the only lag that is non-zero.

Problem 1.4

Suppose that ${e_t}$ is a zero mean white noise process with variance $\sigma^2$. Consider: \begin{enumerate} \item[(i)] $y_t = 0.80y_{t−1} − 0.15y_{t−2} + e_t − 0.30e_{t−1}$ \item[(ii)] $y_t = y_{t−1} − 0.50y_{t−2} + e_t − 1.2e_{t−1}$. \end{enumerate}

Part (a)

\fbox{Identify each model as an ARMA(p, q) process; that is, specify $p$ and $q$.}

\begin{enumerate} \item We rewrite equation (i),

Now, we rewrite it into the form \begin{align*} (1 - 0.8 B + 0.15 B^2) y_t &= (1 - 0.3 B) e_t\ -20 (1 - 0.5 B)(1 - 0.3 B) y_t &= (1 - 0.3 B) e_t\ -20 (1 - 0.5 B) y_t &= e_t. \end{align*}

We see that $y_t = 0.5 y_{t-1} - \frac{e_t}{20}$. Two things should be pointed out. First, assuming $e_t$ is symmetric with zero mean, $- \frac{e_t}{20}$ is distributed the same as $\frac{e_t}{20}$. Next, the variance of $\frac{e_t}{20}$ is $\frac{1}{400} \sigma^2$.

We let $W_t = \frac{1}{20} e_t$, and thus

where ${W_t}$ is a zero mean white noise process with variance $\frac{1}{400} \sigma^2$ and ${y_t}$ is AR(1).

\item We rewrite equation (ii),

Now, we rewrite it into the form \begin{align*} (1 - B + 0.5 B^2) y_t &= (1 - 1.2 B) e_t\ 0.5 (B - 1 + i)(B - 1 - i) y_t &= (1 - 1.2 B) e_t. \end{align*} We see that this is an ARMA(2,1) process. \end{enumerate}

Part (b)

\fbox{Determine whether each model is stationary and/or invertible.} Time series (i) is AR(1) and is thus invertible. We also know that it is stationary since $|φ| = |0.5| < 1$.

Time series (ii) is ARMA(2,1). Let $φ(x) = (x - 1 + i)(x - 1 - i)$ which has roots $1+i$ and $1-i$, which both modulus $\sqrt{2}$. This is larger than $1$, so it is invertible. Let $\theta(x) = 1 - 1.2 x$ which has root $0.8\overbar{3}$. Since $|0.8\overbar{3}| < 1$, it is not stationary.

Problem 2.1

The Johnson and Johnson dataset contains quarterly earnings per share for the U.S. company Johnson & Johnson. There are 84 quarters (21 years) measured from the first quarter of 1960 to the last quarter of 1980. To load the dataset, run the following: install.packages(”astsa”); library(astsa). The dataset is under the name jj. Do a log transformation of the original time series before answering the following.

Preliminary analysis

We would like to take a look at a simple plot of the data, prior to any transformations.

library(astsa)
tsdata <- ts(data=jj)
plot(tsdata)

We see that the variance increases over time. The log-transformation will fix this problem, as computed in the following code:

n <- length(tsdata)
A <- exp((1/n)*sum(log(tsdata)))
ys <- A*log(tsdata)
log_j <- log(tsdata)

Part (a)

\fbox{\begin{minipage}{.8\textwidth} Construct a time series plot for the logged data. Comment on overall trend and seasonality variation. \end{minipage}}

We generate the plot with the following R code:

plot(ys)

The data has both seasonality and a (positive) trend.

Part (b)

\fbox{\begin{minipage}{.8\textwidth} Fit the a regression model on the logged data

where $Q_i(t) = 1$ if time $t$ corresponds to quarter $i = 1, 2, 3$ and zero otherwise. Assume $e_t$ is a normal white noise sequence. Report model coefficients estimates. Superimpose the fitted values on the time plot in part (a). Note: you will need to first create a variable for time and quarter. To do that, you may use: t=1:84; qt=as.factor(rep(1:4,21)). \end{minipage}}

We perform the model fitting using the following R code:

t <- 1:n
qt <- as.factor(rep(1:4,(n/4)))
q1 <- qt==1
q2 <- qt==2
q3 <- qt==3
m <- cbind(t,q1,q2,q3,ys)

# fit regression model to data
fit <- lm(ys~t+q1+q2+q3, data=m)
fit2 <- lm(log_j~t+q1+q2+q3, data=m)



qt2 <- as.factor(rep(1:4,(n/4)))
fit3 <- lm(log_j~t+qt)

# better approach:
#    fit <- lm(ys~t+qt)
# where qt are the factors (1,2,3,4)

The model coefficients are given by:

summary(fit)

## 
## Call:
## lm(formula = ys ~ t + q1 + q2 + q3, data = m)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.8847 -0.2735 -0.0356  0.2553  0.8342 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.508319   0.111529 -22.490  < 2e-16 ***
## t            0.126112   0.001704  73.999  < 2e-16 ***
## q1           0.514570   0.116866   4.403 3.31e-05 ***
## q2           0.599431   0.116803   5.132 2.01e-06 ***
## q3           0.810985   0.116766   6.945 9.50e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3783 on 79 degrees of freedom
## Multiple R-squared:  0.9859,	Adjusted R-squared:  0.9852 
## F-statistic:  1379 on 4 and 79 DF,  p-value: < 2.2e-16

summary(fit2)

## 
## Call:
## lm(formula = log_j ~ t + q1 + q2 + q3, data = m)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.29318 -0.09062 -0.01180  0.08460  0.27644 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.8312482  0.0369603 -22.490  < 2e-16 ***
## t            0.0417930  0.0005648  73.999  < 2e-16 ***
## q1           0.1705267  0.0387289   4.403 3.31e-05 ***
## q2           0.1986494  0.0387083   5.132 2.01e-06 ***
## q3           0.2687577  0.0386959   6.945 9.50e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1254 on 79 degrees of freedom
## Multiple R-squared:  0.9859,	Adjusted R-squared:  0.9852 
## F-statistic:  1379 on 4 and 79 DF,  p-value: < 2.2e-16

summary(fit3)

## 
## Call:
## lm(formula = log_j ~ t + qt)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.29318 -0.09062 -0.01180  0.08460  0.27644 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.6607215  0.0358430 -18.434  < 2e-16 ***
## t            0.0417930  0.0005648  73.999  < 2e-16 ***
## qt2          0.0281227  0.0386959   0.727   0.4695    
## qt3          0.0982310  0.0387083   2.538   0.0131 *  
## qt4         -0.1705267  0.0387289  -4.403 3.31e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.1254 on 79 degrees of freedom
## Multiple R-squared:  0.9859,	Adjusted R-squared:  0.9852 
## F-statistic:  1379 on 4 and 79 DF,  p-value: < 2.2e-16

In other words, the estimate is given by

The plot of the data with $\hat{y}_t$ superimosed onto it is given by: