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Statistics Indian Statistical Service Part 3 Question-8 Solved Solution






8.(a) (1) Define a moving average process of order q(MA(q) process) and derive its autocorrelation function. A moving average process of order q, denoted as MA(q), is a stationary time series that can be expressed as a linear combination of current and past white noise error terms. Specifically, an MA(q) process can be written as: y_t = μ + ε_t + θ_1 ε_(t-1) + θ_2 ε_(t-2) + ... + θ_q ε_(t-q) where y_t is the current value of the process, μ is the mean of the process, ε_t is the current white noise error term with zero mean and constant variance, and θ_1, θ_2, ..., θ_q are the parameters of the MA(q) process that represent the weights on the past error terms. To derive the autocorrelation function of an MA(q) process, we need to find the correlation between y_t and y_(t-k) for k = 1, 2, ..., q. Since the MA(q) process depends on past error terms, we can use the fact that the error terms are uncorrelated to simplify the calculation of the autocorrelation function. For k = 1, we have: Corr(y_t, y_(t-1)) = Corr(ε_t + θ_1 ε_(t-1) + θ_2 ε_(t-2) + ... + θ_q ε_(t-q), ε_(t-1) + θ_1 ε_(t-2) + θ_2 ε_(t-3) + ... + θ_q ε_(t-q-1)) = Corr(ε_t, ε_(t-1)) + θ_1 Corr(ε_(t-1), ε_(t-1)) + 0 + ... + 0 = 0 + θ_1 * 1 + 0 + ... + 0 = θ_1 For k > 1, we have: Corr(y_t, y_(t-k)) = Corr(ε_t + θ_1 ε_(t-1) + θ_2 ε_(t-2) + ... + θ_q ε_(t-q), ε_(t-k) + θ_1 ε_(t-k+1) + θ_2 ε_(t-k+2) + ... + θ_q ε_(t-q-k+1)) = 0 + 0 + 0 + ... + θ_k * 1 + 0 + ... + 0 = θ_k Therefore, the autocorrelation function of an MA(q) process is given by: ρ_k = 0 for k > q ρ_k = θ_k for k ≤ q where ρ_k is the autocorrelation function at lag k, and θ_k are the parameters of the MA(q) process. This means that the autocorrelation function of an MA(q) process is zero for lags greater than q, and the autocorrelation function for lags up to q depends on the parameters θ_1, θ_2, ..., θ_q. (ii) Stating the underlying conditions, write an MA(1) process as an autoregressive • process of infinite order. An MA(1) process can be written as an autoregressive process of infinite order under the following conditions: Consider an MA(1) process given by: y_t = c + ε_t + θε_{t-1} where ε_t is white noise with zero mean and constant variance, and θ is a constant coefficient. To write this as an autoregressive process of infinite order, we can use the lag operator L such that L^ky_t = y_{t-k}. Then, we can express the MA(1) process as: y_t = c + ε_t + θε_{t-1} y_t = c + ε_t + θLε_t y_t = c + (1+θL)ε_t Now, we can use the formula for the geometric series: 1 / (1 + θL) = 1 - θL + θ^2L^2 - θ^3L^3 + ... Thus, we can express the MA(1) process as an autoregressive process of infinite order: y_t = c + (1+θL)ε_t y_t = c + (1-θL+θ^2L^2-θ^3L^3+...)ε_t y_t = c + ε_t - θε_{t-1} + θ^2ε_{t-2} - θ^3ε_{t-3} + ... Therefore, an MA(1) process can be written as an autoregressive process of infinite order with coefficients {1, -θ, θ^2, -θ^3, ...}. (iii) How can we identify and select the order of autoregressive and moving average processes. Identifying and selecting the appropriate order of autoregressive (AR) and moving average (MA) processes is an important step in time series analysis. Here are some commonly used methods to determine the order of an AR or MA process: Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF): The ACF and PACF are commonly used to identify the order of an AR or MA process. For an AR process, the ACF will have a slow decay and the PACF will have a sharp cutoff after the lag corresponding to the AR order. For an MA process, the PACF will have a slow decay and the ACF will have a sharp cutoff after the lag corresponding to the MA order. Information Criteria: Information criteria such as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) can be used to select the order of an AR or MA process. These criteria balance the goodness of fit with the complexity of the model, and the model with the lowest AIC or BIC is considered the best. Box-Jenkins Methodology: The Box-Jenkins methodology involves a systematic approach to identifying and selecting the order of an AR or MA process. It involves model identification, estimation, and diagnostic checking. The order of the AR and MA terms is determined using the ACF and PACF, and the model is estimated using maximum likelihood estimation. The residuals of the model are then checked for autocorrelation and other patterns using diagnostic plots. Expert Knowledge: In some cases, expert knowledge of the underlying process can be used to select the order of an AR or MA process. For example, if the process is known to be seasonal, a seasonal AR or MA term may be included in the model. Overall, selecting the appropriate order of an AR or MA process is important for obtaining accurate forecasts and understanding the underlying process. Different methods can be used in combination to determine the best order for a particular time series.


8.(b) (i) Define spectral density function and normalized form of spectral density function of a stationary process. Prove that the normalized spectral density function is a Fourier transform of autocorrelation function (ACF). The spectral density function is a tool used in the analysis of stationary stochastic processes. It characterizes the distribution of the process's power over different frequencies. The spectral density function is a function of frequency and is denoted by $f(\omega)$ or $S(\omega)$. The normalized form of the spectral density function is defined as: $$\phi(\omega) = \frac{S(\omega)}{2\pi}$$ where $2\pi$ is a normalization constant chosen so that the area under the spectral density function equals the variance of the process. Now, to prove that the normalized spectral density function is the Fourier transform of the autocorrelation function (ACF), we can use the following steps: 1. Start with the definition of the ACF for a stationary process $X_t$: $$\gamma(h) = \operatorname{Cov}(X_t, X_{t+h})$$ 2. Take the Fourier transform of the ACF with respect to the lag $h$, i.e., $$\hat{\gamma}(\omega) = \sum_{h=-\infty}^{\infty} \gamma(h) e^{-i\omega h}$$ where $\hat{\gamma}(\omega)$ is the Fourier transform of the ACF. 3. Using the definition of the spectral density function, we have $$S(\omega) = \sum_{h=-\infty}^{\infty} \gamma(h) e^{-i\omega h}$$ 4. Divide both sides by $2\pi$ to obtain the normalized spectral density function: $$\phi(\omega) = \frac{1}{2\pi} \sum_{h=-\infty}^{\infty} \gamma(h) e^{-i\omega h}$$ 5. Finally, recognize that the right-hand side of the above equation is the inverse Fourier transform of $\hat{\gamma}(\omega)$. Thus, we have $$\phi(\omega) = \frac{1}{2\pi} \hat{\gamma}(-\omega) = \frac{1}{2\pi} \hat{\gamma}(\omega)$$ where the last equality follows from the symmetry of the Fourier transform for real-valued functions. Therefore, we have shown that the normalized spectral density function is the Fourier transform of the autocorrelation function for a stationary process. This result is important in the analysis of time series data as it allows us to obtain information about the frequency content of the process from its autocorrelation function. (ii) Derive the spectral density function of an AR (1) process. To derive the spectral density function of an AR(1) process, we start by expressing the process as a linear combination of its own past values and a white noise error term: $$y_t = \phi y_{t-1} + \epsilon_t$$ where $\phi$ is the coefficient of the first lag, and $\epsilon_t$ is a white noise error term with variance $\sigma_{\epsilon}^2$ and mean zero. To find the spectral density function, we first find the autocorrelation function (ACF) of the process. We start by taking the expected value of both sides of the equation: $$E(y_t) = E(\phi y_{t-1} + \epsilon_t)$$ Since the process is stationary, $E(y_t)$ is a constant, which we can denote as $\mu$. Since the error term has mean zero, $E(\epsilon_t) = 0$. We can then rearrange the equation to get: $$y_{t-1} = \frac{1}{\phi} (y_t - \epsilon_t)$$ Substituting this expression into the original equation, we get: $$y_t = \phi \frac{1}{\phi} (y_t - \epsilon_t) + \epsilon_t = y_t - \phi \epsilon_t + \epsilon_t$$ Simplifying, we get: $$(1-\phi L) y_t = \epsilon_t$$ where $L$ is the lag operator, such that $L^k y_t = y_{t-k}$. Solving for $y_t$, we get: $$y_t = \frac{1}{1-\phi L} \epsilon_t$$ We can then express the spectral density function $f(\lambda)$ as the Fourier transform of the ACF, which we can find by taking the covariance between $y_t$ and $y_{t-k}$, where $k$ is the lag: $$\text{cov}(y_t, y_{t-k}) = E(y_t y_{t-k}) - E(y_t)E(y_{t-k})$$ Since the process is stationary, $E(y_t)$ and $E(y_{t-k})$ are both equal to $\mu$. We can also use the expression for $y_t$ in terms of the error term to get: $$\text{cov}(y_t, y_{t-k}) = E\left(\frac{1}{1-\phi L} \epsilon_t \frac{1}{1-\phi L} \epsilon_{t-k}\right) = \frac{1}{1-\phi^2} \sigma_{\epsilon}^2 \phi^k$$ where we have used the fact that the ACF of a white noise process is zero for all lags. We can then express the spectral density function as: $$f(\lambda) = \sum_{k=-\infty}^{\infty} \text{cov}(y_t, y_{t-k}) e^{-i\lambda k}$$ Substituting the expression for the ACF, we get: $$f(\lambda) = \frac{\sigma_{\epsilon}^2}{2\pi} \frac{1}{1-2\phi\cos(\lambda) + \phi^2}$$ This is the spectral density function of an AR(1) process.

8.(c) (i) Establish the relationship between periodogram and sample autocovariance function (ACF).

The periodogram is a non-parametric estimator of the spectral density function, whereas the ACF is a measure of linear dependence between two observations of a time series at different time lags. The periodogram can be computed by taking the absolute squared value of the discrete Fourier transform of a time series. The discrete Fourier transform converts a time series from the time domain to the frequency domain. Specifically, for a time series $x_1, x_2, \ldots, x_n$, the periodogram is given by: $$I(\omega) = \frac{1}{n} \left|\sum_{t=1}^n x_t e^{-i \omega t} \right|^2$$ where $\omega$ is the frequency in radians per time unit. On the other hand, the ACF measures the linear dependence between two observations of a time series at different time lags. The ACF at lag $k$ is defined as: $$\hat{\rho_k} = \frac{\sum_{t=k+1}^n (x_t - \bar{x})(x_{t-k} - \bar{x})}{\sum_{t=1}^n (x_t - \bar{x})^2}$$ where $\bar{x}$ is the sample mean of the time series. The relationship between the periodogram and ACF is established by the Wiener-Khinchin theorem, which states that the spectral density function is the Fourier transform of the ACF. In other words, if $f(\cdot)$ denotes the Fourier transform operator, then the normalized spectral density function $S(\omega)$ of a stationary time series $x_1, x_2, \ldots, x_n$ is given by: $$S(\omega) = f{\rho(k)}$$ where $\rho(k)$ is the ACF at lag $k$. Therefore, the periodogram can be seen as an estimate of the spectral density function, which is proportional to the squared modulus of the Fourier transform of the ACF.

(ii) Show that the periodogram provides an estimate of the spectral density function. What kind of preprocessing of data is required before performing periodigram analysis? The periodogram is a commonly used tool in spectral analysis, which provides an estimate of the spectral density function of a stationary time series. The spectral density function measures the distribution of the power of the signal across different frequencies, and it is an important tool for understanding the underlying dynamics of a time series. The periodogram is obtained by computing the discrete Fourier transform of the time series, and then taking the squared magnitude of the resulting complex-valued sequence. Specifically, if $x_1, x_2, \ldots, x_n$ is a stationary time series, the periodogram is given by: $$I(\omega) = \frac{1}{n} \left| \sum_{t=1}^n x_t e^{-i\omega t} \right|^2$$ where $\omega$ is the frequency in radians per time unit. Under certain conditions, it can be shown that the periodogram is a consistent estimator of the true spectral density function of the time series. That is, as the length of the time series goes to infinity, the periodogram converges to the true spectral density function in probability. However, to obtain reliable estimates of the spectral density function using the periodogram, it is important to preprocess the data appropriately. In particular, the following steps are recommended: Detrend the data: Remove any linear trend from the time series, as this can distort the frequency content of the signal. Apply a window function: Multiply the time series by a window function, which tapers the edges of the data to reduce spectral leakage. Popular window functions include the Hanning, Hamming, and Blackman windows. Zero-pad the data: Add zeros to the end of the time series to increase the resolution of the frequency domain representation. The length of the zero-padding should be chosen to balance the trade-off between frequency resolution and computational efficiency. By performing these preprocessing steps, the periodogram can provide reliable estimates of the spectral density function of a stationary time series.



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