class: center, middle, inverse, title-slide # Introduction ### Kevin Kotzé --- <!-- layout: true --> <!-- background-image: url(image/logo.svg) --> <!-- background-position: 2% 98% --> <!-- background-size: 10% --> --- # Contents 1. Overview 1. Introduction 1. Decomposing a time series 1. Popular Processes 1. Difference Equations & Lag Operators 1. Conditional & Unconditional Moments 1. Stationarity, autocorrelation & ergodicity 1. Impact multipliers & IRFs 1. Conclusion --- # Time Series Approaches - Time Domain - modelling current on past (i.e. reduced-form) - Box-Jenkins method - State-space method - Frequency domain - modelling period or systematic variations - Includes, Fourier analysis, power spectra's & wavelet transforms - These methodologies may be applied to VAR, SVAR, CVAR, GARCH, MVGARCH, SV, DFM, MSW, STAR, `\(\dots\)` --- # Time Series Approaches - Consider the simple linear regression model `\begin{eqnarray} y_{t} = \underbrace{x_t^{\top} \beta}_\text{explained} + \underbrace{\varepsilon_{t}}_\text{unexplained}\; , & \hspace{1cm} & t=1, \ldots , T \end{eqnarray}` - Errors should not be serially correlated for least squares estimates in such a model: - `\(\mathbb{E}[\varepsilon_t] = \mathbb{E}[\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2},\dots] = 0\)` and - `\(\mathbb{E}\left[\varepsilon_i \varepsilon_{t-j}\right] = 0\)`, for `\(j \ne 0\)` - If there is serial correlation in the errors then estimates are inefficient --- # Introduction - Most economic and financial time series exhibit some form of serial correlation - If economic output is large during the previous quarter then there is a good chance that it is going to be large in the current quarter - A change that arises in this quarter may only impact on other variables at a distant future point - Particular shock may affect variables over successive quarters - Hence, we need to start thinking about the dynamic structure of the system that we are investigating - Most time series models look at explicitly allowing for these features, while adhering to the statistical properties mentioned above --- # Introduction - Modern day time series is concerned with interpreting data that is measured at discrete intervals - Traditionally, a large part of time series analysis is concerned with forecasting - Testing various hypotheses (theories) that may be used to describe the past behaviour of a time series variable - These objectives are best achieved by identifying the *dynamic path* of a time series - To do this we need to decompose the series into the constituent components --- background-image: url(image/pic1.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 1: Decomposition of Time Series --- # Basic time series model - Consider the following example: `\begin{eqnarray} Trend:& & T_t = 1+0.1t \\ Seasonal:& & S_t = 1.6 \sin(t \pi /6) \\ Irregular:& & I_t = 0.7 I_{t-1} + \varepsilon_t \end{eqnarray}` where `\(\varepsilon_t \;\)` is a random disturbance - The trend component is deterministic - The seasonal is also deterministic and uses a sinusoidal to impart cyclical behaviour - The irregular component contains a stochastic term, which may be described by a statistical distribution --- background-image: url(image/SA_data.svg) background-position: top background-size: 85% 85% class: clear, center, bottom Figure 2: Real World Data: South African Data --- # Economic data - Consider the information content that is contained in the series - Most economic data contains trends, seasonals, and irregular components - Most economic data is measured in discrete time (with relatively long intervals) - Some financial data is measured at extremely high frequency - This data may be expressed as rates, indices or totals - Be cautions of using interpolation & moving to lower frequency - Many economic variables are subject to revision --- # Economic data - Take note of the frequency and the type of transformation that could (should) be applied - Common transformations include: calculation of growth rates `\([\log(GDP_{t}/GDP_{t-1})]\)`, the calculation of annualised rates `\([(1+(i_t/100))^{(1/12)}-1]\)`, etc. - Most countries follow globally accepted measurement practices --- # Financial data - Data on stock prices and indices is overwhelming: - Does the data contain true trading prices, quotes, or proxies for trading prices? - May only be interested in buyer initiated (ask) or seller initiated (bid) orders? - Do the prices include transaction costs, commissions & effects of tax transfers? - Is the market sufficiently liquid? - Have the prices been adjusted for inflation, or have they been correctly discounted? --- # Financial data - At what frequency do you want to measure trading activity/returns? - Is it feasible to use extremely high frequency data and what does it represent? - Consider the implications of limiting data to a particular sub-sample - Transformation to returns generally displays more stationary behaviour - Represents a complete scale-free summary about an investment opportunity --- # Processes - Time series is a collection of observations indexed by the date of each realisation - Using notation that starts at time, `\(t = 1\)` and using the end point, `\(t =T\)`, `\begin{eqnarray} \left\{ y_{1}, y_{2}, y_{3}, \ldots , y_{T}\right\} \end{eqnarray}` - Time index can be of any frequency (e.g. daily, quarterly, etc.) --- # Deterministic & Stochastic processes - Deterministic process will always produce the same output from a given starting condition or initial state - No element of randomness, i.e. `\(T_t = 1+0.1t\)` - Stochastic process has some indeterminacy that relates to the future evolution of the process - Usually described by some form of statistical distribution - Examples include: white noise processes, random walks, Brownian motions, Markov chains, martingale difference sequences, etc. --- # Stochastic processes: White noise - Serially uncorrelated random variables with zero mean and finite variance - Errors may follow a normal distribution - Gaussian white noise process - Slightly stronger condition is that they are independent from one another `\begin{eqnarray} \varepsilon_t \sim \mathsf{i.i.d.} & \mathcal{N}(0, \sigma_{\varepsilon_t}^2) \end{eqnarray}` - Notice three implications of this assumption: - `\(\mathbb{E}[\varepsilon_t] = \mathbb{E}[\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-1}, \dots ] =0\)` - `\(\mathbb{E}[\varepsilon_t \varepsilon_{t-j}] = \mathsf{cov}[\varepsilon_{t} \varepsilon_{t-j}] = 0\)` - `\(\mathsf{var}[\varepsilon_{t}] = \mathsf{cov}[\varepsilon_{t}\varepsilon_{t}] = \sigma_{\varepsilon_t}^2\)` --- background-image: url(image/wn.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 3: Gaussian White Noise Process <!-- # White Noise Process --> <!-- No persistence -> looks like noise --> --- # Random Walk - Random walk would imply that the effect of a shock is permanent `\begin{eqnarray} y_t = y_{t-1} + \varepsilon_t \end{eqnarray}` - Could be represented as `\begin{eqnarray} y_t = \sum_{j=1}^{t} \varepsilon_j \end{eqnarray}` - Shocks have permanent effects --- background-image: url(image/rw.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 4: Random Walk - Simulated Time Series <!-- # Random Walk --> --- background-image: url(image/rw_shock.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 5: Random Walk - Effect of Shock [ `\(y_{-1}=0, \varepsilon_0 = 1\)` and `\((\varepsilon_1, \dots) = 0\)` ] <!-- # Random Walk --> --- # Random Walk plus Drift - Random walk plus a constant term `\begin{eqnarray} y_t = \mu + y_{t-1} + \varepsilon_t \end{eqnarray}` - This could be represented as `\begin{eqnarray} y_t = \mu \cdot t + \sum_{j=1}^{t} \varepsilon_j \end{eqnarray}` - Shocks have permanent effects and are influenced by the drift --- background-image: url(image/rwd.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 6: Random Walk plus Drift - Simulated Time Series [Dotted: `\(\mu=1.2\)` & Solid: `\(\mu = 0.5\)`] <!-- # Random Walk with Drift --> --- background-image: url(image/rwd_shock.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 7: Random Walk with Drift - Effect of Shock [ `\(y_{-1} = 0\)`, `\(\mu = 1.2\)`, `\(\varepsilon_0 = 1\)` and `\((\varepsilon_1, \ldots) = 0\)`] --- background-image: url(image/rw_m.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 8: Different Random Walks --- # Autoregressive process - An `\(AR(1)\)` process describes situations where the present value of a time series is a linear function of the previous observation `\begin{eqnarray} y_{t}= \phi y_{t-1} + \varepsilon_{t} & \; & \varepsilon_t \sim \mathsf{i.i.d.} \; \mathcal{N}(0, \sigma_{\varepsilon_t}^2) \end{eqnarray}` - Know something about the *conditional* distribution of `\(y_t\)` given `\(y_{t-1}\)` - After repeated substitution it would take the form `\begin{eqnarray} y_t = \phi^j \sum^t_{j=1} \varepsilon_j \end{eqnarray}` - Could include several lags, `\(AR(p)\)` model and the distribution of the error term could take many forms - Think about the implication of future values of `\(y_t\)` when `\(\phi=0.5,1,\)` or `\(1.5\)`? - Think about the role of the constant in a random walk plus drift and in a stationary `\(AR(1)\)`? --- background-image: url(image/ar.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 9: `\(AR(1)\)` - Simulated Time Series [ `\(\phi = 0.9\)`] <!-- # Autoregressive process --> --- background-image: url(image/ar_shock.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 10: `\(AR(1)\)` - Effect of Shock [ `\(\phi = 0.9\)`] --- # Moving Average process - `\(MA(q)\)` model describes a time series by the weighted sum of the current and previous errors - Consider examples where it takes a bit of time for the error (or "shock") to dissipate `\begin{eqnarray} y_t = \varepsilon_t + \theta \varepsilon_{t-1} \end{eqnarray}` - This type of expression may be used to describe a wide variety of stationary time series processes --- background-image: url(image/ma.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 11: `\(MA(1)\)` - Simulated Time Series [ `\(\theta = 0.7\)`] --- background-image: url(image/ma_shock.svg) background-position: top background-size: 90% 90% class: clear, center, bottom Figure 12: `\(MA(1)\)` - Effect of Shock [ `\(\theta = 0.7\)`] --- # ARMA process - A combination of these modes is termed an `\(ARMA(1,1)\)` `\begin{eqnarray} y_t = \phi y_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1} \end{eqnarray}` - where an `\(ARMA(p, q)\)` takes the form `\begin{eqnarray} y_t &=& \phi_1 y_{t-1} + \phi_2 y_{t-2} + ... + \phi_p y_{t-p} + \ldots \\ && \ldots + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \ldots + \theta_q \varepsilon_{t-q} \end{eqnarray}` - This model was popularized by Box & Jenkins, who developed a methodology that may be used to identify the terms that should be included in the model - Note that the `\(AR(p)\)`, `\(MA(q)\)` and `\(ARMA(p,q)\)` may provide some form of characterisation of the South Africa data that we saw previously --- # Long Memory & Fractional Differencing - Most `\(AR(p)\)`, `\(MA(q)\)` and `\(ARMA(p,q)\)` processes are often referred to as a short-memory process because the coefficients in the representation are dominated by exponential decay - Long memory (or persistent) time series are considered intermediate compromises between the short memory models and integrated nonstationary processes - Long periods during which observations tend to be at a high level and similar long periods during which observations tend to be at a low level --- # Difference Equations - Linear first order difference equation may be expressed as `\begin{eqnarray} \ y_{t}=\phi y_{t-1}+\varepsilon _{t}\qquad \end{eqnarray}` - Relate `\({y}\)` at time `\(t\)`, to its previous value in time `\((t-1)\)` - If `\(|\phi| <1\)`, we can show that the series will always return to its mean after a shock - If `\(\phi =1\)` then the difference equation is a random walk - Higher Order Difference Equations may include `\begin{eqnarray} y_{t}=\phi_{1} y_{t-1}+ \phi_{2} y_{t-2}+\varepsilon _{t} \end{eqnarray}` --- # Lag Operators - Convenient tools to analyse difference equations `\begin{equation} \ Ly_{t}=y_{t-1} \label{lag1} \end{equation}` - Similarly, `\(L^{-1}y_{t}=y_{t+1}\)` - Can be raised to arbitrary integer powers `\(k\)` such that `\begin{eqnarray} L^{k}y_{t}=y_{t-k} \\ L^{-k}y_{t}=y_{t+k} \label{lag_k} \end{eqnarray}` --- # Lag Operators - The first difference of a series could then be written as `\begin{equation} \ \left( 1-L\right) y_{t}=y_{t}-y_{t-1} \label{1st_dif} \end{equation}` - The four-period difference would be defined as `\begin{equation} \left( 1-L^{4}\right) y_{t}=y_{t}-y_{4} \end{equation}` - The second difference of a time series is then `\begin{equation} \left( 1-L\right) ^{2}y_{t} = \left( 1-L\right) y_{t} - \left( 1-L\right) y_{t-1} = (y_{t}-y_{t-1}) - (y_{t-1}-y_{t-2}) \label{2nd_dif} \end{equation}` --- # Lag Operators - Higher order expressions make use of a polynomial of lag operators `\begin{equation} \phi (L)=\left( 1-\phi _{1}L-\phi _{2}L^{2}- \ldots - \phi _{p}L^{p}\right) \end{equation}` - where `\(\phi\)` is a vector of coefficients - When applying a particular lag polynomial to a time series `\(y_{t}\)`, we could use the expression `\begin{eqnarray} \phi (L)y_{t} &\equiv &\left( 1-\overset{p}{\underset{i=1}{\sum }}\phi_{i}L^{i}\right) y_{t} \\ &=&\left( 1-\phi _{1}L-\phi _{2}L^{2}- \ldots - \phi _{p}L^{p}\right) y_{t} \\ &=&y_{t}-\phi _{1}y_{t-1}-\phi _{2}y_{t-2} - \ldots -\phi _{p}y_{t-p} \end{eqnarray}` --- # Moments of Distribution - Distributions often summarised by first (mean) and second (variance) moments - Higher order moments may be of interest (skewness & kurtosis) - Always important to distinguish between the unconditional and conditional distributions --- # Moments of Distribution - The first moment of a stochastic process is the average of `\(y_{t}\)` over all possible realisations `\begin{eqnarray} \bar{y} =\mathbb{E}\left[ y_{t}\right], \;\;\;\; t=1, \dots , T \end{eqnarray}` - The second moment is defined as the variance `\begin{eqnarray} \mathsf{var}[y_{t}]=\mathbb{E}\left\{ y_{t}\;y_{t}\right\} =\mathbb{E}\left\{ \left(y_{t}-\mathbb{E}\left[y_{t}\right]\right)^{2}\right\}, \;\;\;\; t=1, \dots , T \end{eqnarray}` - And the covariance, for `\(j\)`: `\begin{eqnarray} \mathsf{cov}[y_{t},y_{t-j}]&=& \mathbb{E}\left\{ y_{t}\;y_{t-j}\right\} \\ &=& \mathbb{E}\left\{\left(y_{t}-\mathbb{E}\left[y_{t}\right]\right) \left(y_{t-j}- \mathbb{E}\left[y_{t-j}\right]\right)\right\} , \;\; t=j+1, \dots , T \end{eqnarray}` --- # Conditional Moments - Conditional distribution is based on past realisations of a random variable - For the `\(AR(1)\)` model `\begin{equation} \ y_{t}=\phi y_{t-1}+\varepsilon _{t} \end{equation}` - where `\(\varepsilon _{t}\sim \mathsf{i.i.d.} \mathcal{N}(0,\sigma ^{2})\)` is Gaussian white noise and `\(|\phi |{<}1\)` - Conditional moments satisfy `\begin{eqnarray} &&\mathbb{E}\left[ y_{t}|y_{t-1}\right] =\phi y_{t-1} \\ &&\mathsf{var}[y_{t}|y_{t-1}] =\mathbb{E}[\phi y_{t-1}+\varepsilon _{t}-\phi y_{t-1}]^{2}=\mathbb{E}[\varepsilon _{t}]^{2}=\sigma ^{2} \\ &&\mathsf{cov}\left[\left(y_{t}|y_{t-1}\right),\left(y_{t-j}|y_{t-j-1}\right)\right] =0 \;\; \text{for } j >1 \label{cond_m} \end{eqnarray}` --- # Conditional Moments - Conditioning on `\(y_{t-2}\)` for `\(y_t\)` `\begin{eqnarray} \mathbb{E}\left[ y_{t}|y_{t-2}\right] &=&\phi ^{2}y_{t-2} \\ \mathsf{var}[y_{t}|y_{t-2}] &=&(1+\phi ^{2})\sigma ^{2} \\ \mathsf{cov}\left[\left(y_{t}|y_{t-2}\right),\left(y_{t-j}|y_{t-j-2}\right)\right] &=&\phi \sigma ^{2} \;\; \text{for } j = 1 \\ \mathsf{cov}\left[\left(y_{t}|y_{t-2}\right),\left(y_{t-j}|y_{t-j-2}\right)\right] &=&0 \;\; \text{for } j > 1 %\label{cond_m2} \end{eqnarray}` --- # Unconditional Moments - Unconditional distribution has slightly different moments for the `\(AR(1)\)` model `\begin{equation} \ y_{t}=\phi y_{t-1}+\varepsilon _{t} \end{equation}` - where `\(\varepsilon _{t}\sim \mathsf{i.i.d.} \mathcal{N}(0,\sigma ^{2})\)` is Gaussian white noise and `\(|\phi |{<}1\)` - Unconditional moments satisfy `\begin{eqnarray} \mathbb{E}\left[ y_{t}\right] &=&0 \\ \mathsf{var}[y_{t}] &=&\frac{\sigma ^{2}}{1-\phi } \\ \mathsf{cov}[y_{t}\;y_{t-j}] &=&\phi ^{j}\mathsf{var}(y_{t}) \label{uncon_m} \end{eqnarray}` --- # Stationarity: Strictly stationary - Time series is strictly stationary if for any values `\begin{eqnarray} \{j_{1}, j_{2}, \dots , j_{n}\}, \end{eqnarray}` - the joint distribution of `\begin{eqnarray} \{ y_{t}, y_{t+j,1} , y_{t+j ,2}, \dots , y_{t+j,n} \} \end{eqnarray}` - depend only on the intervals separating the dates `\begin{eqnarray} \{ j_{1}, j_{2}, \dots , j_{n}\} \end{eqnarray}` - and not on the date itself, `\(t\)` --- # Stationarity: Covariance stationary - If neither the mean, `\(\bar{y}\)`, nor the covariance, `\(\mathsf{cov}(y_{t}\;y_{t-j})\)`, depend on the date, `\(t\)` - Then the process for `\(y_{t}\)` is said to be covariance (weakly) stationary, where for all `\(t\)` and any `\(j\)` `\begin{eqnarray} \mathbb{E}\left[ y_{t}\right] &=&\bar{y} \\ \mathbb{E}\left[ \left( y_{t}-\bar{y} \right) \left( y_{t-j}-\bar{y} \right) \right] &=&\mathsf{cov}(y_{t}\;y_{t-j}) \end{eqnarray}` - When referring to stationarity in the remainder of the course we refer to covariance stationarity - Note that the process `\(y_{t}=\alpha t+\varepsilon _{t}\)` would not be stationary, as the mean clearly depends on `\({t}\)` - In addition, we saw that the unconditional moments of the `\(AR(1)\)` with `\(|\phi|<1\)` had a mean and covariance that did not depend on time --- # Autocorrelation function (ACF) - For a stationary process we can plot the covariance of the process against a number of lags - Makes use of the autocovariance function, which is denoted `\(\gamma \left(j\right) \equiv \mathsf{cov}\left( y_{t} \; y_{t-j}\right)\)` for `\(t=1,\ldots , T\)` - Autocovariance function may be standardized by dividing the variance to derive the ACF `\begin{eqnarray} \rho \left(j\right) \equiv \frac{\gamma \left( j\right) }{\gamma \left( 0\right)} \end{eqnarray}` - Useful to make a plot of `\(\rho \left( j\right)\)` against (non-negative) `\(j\)` to learn about the properties of a time series --- # Partial autocorrelation function (PACF) - With an `\(AR(1)\)` process, `\(y_{t}=\phi y_{t-1}+\varepsilon _{t}\)`, the ACF would suggest `\(y_t\)` and `\(y_{t-2}\)` are correlated even though `\(y_{t-2}\)` does not appear in the model - Result of the pass-through, where `\(y_t = \phi^2 y_{t-2}\)` - PACF eliminates effects of intervening values and focuses on relationship between `\(y_t\)` and `\(y_{t-2}\)` - Hence, the PACF `\((y_t,y_{t-j})\)` eliminates the effects of intervening correlations between `\(y_{t-1}\)` and `\(y_{t-j-1}\)` --- # Partial autocorrelation function (PACF) - To construct a PACF one would usually make use of the following steps, - Demean the series `\((y_t^\star = y_t - \bar{y})\)` - Form the `\(AR(1)\)` equation, `\(y_t^\star = \phi_{11} y_{t-1}^\star + \upsilon_t\)`, where `\(\upsilon_t\)` may not be white noise. In this case, `\(\phi_{11}\)` is `\(\rho(1)\)` in the ACF. Also equal to the first coefficient in the PACF as there are no intervening values between `\(y_t\)` and `\(y_{t-1}\)` - Now form the second-order autoregression, `\(y_t^\star = \phi_{21} y_{t-1}^\star + \phi_{22} y_{t-2}^\star + \upsilon_t\)`. In this case, `\(\phi_{22}\)` is the PACF between `\(y_t\)` and `\(y_{t-2}\)`, since the effects of `\(y_{t-1}^\star\)` on `\(y_{t}^\star\)` are captured by the `\(\phi_{21}\)`, which isolate the effects of `\(y_{t-1}\)` on `\(y_t\)` - etc. --- # Q-statistic - Box-Ljung `\(Q\)`-statistic tests whether series is white noise - Tests whether a group of autocorrelations differ from zero - Tests the "overall" randomness based on a number of lags `\begin{eqnarray} Q(k) = T(T+2) \sum_{j=1}^{k} \frac{\rho_j^2}{T-j} \end{eqnarray}` - where `\(\rho\)` refers to the residual autocorrelation from lag `\(j\)` - In this case we would express `\(\rho\)` as `\begin{eqnarray} \rho_k = \frac{\sum_{t=1}^{T-k} (\varepsilon_t - \bar{\varepsilon})(\varepsilon_{t+k} - \bar{\varepsilon} )}{\sum_{t=1}^{T} (\varepsilon_t - \bar{\varepsilon})^2} \end{eqnarray}` - where `\(\bar{\varepsilon}\)` is the mean of the `\(T\)` residuals --- background-image: url(image/q_stat1.svg) background-position: top background-size: 80% 80% class: clear, center, bottom Figure : Serial Correlation --- # Ergodicity - Covariance-stationary process is ergodic in the mean when the sample average, `\(\bar{y}\)` `\(\equiv \left( 1/T\right) \sum^{T}_{t=1} y_{t}\)`, converges in probability to the population `\(\mathbb{E}\left[ y_{t}\right]\)` as `\(T\rightarrow\)` `\(\infty\)` - Similar statement could be made for a process that is ergodic in the variance (or autocovariance) - When ergodicity holds, the sample average and variance provide a consistent estimate of their population counterparts --- # Impact multipliers - May investigate cause & effects of events - Estimate the response of GDP growth after unexpected `\(1\)`% increase in demand - How will the exchange rate react to an unexpected increase in the interest rate - Assuming stationarity, any `\(AR(p)\)` process can be written as an infinite order MA (later in course) - Implies that an `\(AR(1)\)` process may be written as `\begin{equation} y_{t}=\varepsilon _{t}+\phi \varepsilon _{t-1}+\phi ^{2}\varepsilon_{t-2}+ \dots = \overset{\infty }{\underset{j=0}{\sum }}\phi^{j}\varepsilon_{t-j} \end{equation}` - Suggests `\(y_{t}\)` can be described by past & present errors / shocks --- # Impact multipliers - Assume the dynamic simulation started at time `\(j\)`, taking `\(y_{t-(j+1)}\)` as given - Effect of a change in the initial shock on `\(y_{t}\)` is then `\begin{equation} \frac{\partial y_{t}}{\partial \varepsilon_{t-j}}=\phi ^{j} \end{equation}` - Termed the dynamic multiplier that depends only on `\(j\)`, `\(\varepsilon _{t-j}\)`, and `\(y_{t}\)` - This expression does not depend on `\(t\)`, (date of observation) --- # Impulse response functions (IRF) - Cumulative effect of temporary shock is then `\begin{equation} \sum_{j=0}^{\infty} \frac{\partial y_{t}}{\partial \varepsilon_{t-j}} = 1+\phi +\phi ^{2}+ \dots +\phi^{j}=\frac{1}{\left( 1-\phi\right)} \end{equation}` - Different values of `\(\phi\)` produce a variety of responses in `\(y_{t}\)` - When `\(|\phi| < 1\)`, the process decays geometrically towards zero - When `\(0<\phi<1\)`, there will be a smooth decay - When `\(0>\phi>-1\)` there will be an oscillating decay - We say that a system described in this way is stable - Dynamic multipliers can be moved forward in time, such that `\(\frac{\partial y_{t+j}}{\partial \varepsilon _{t}}= \phi ^{j}\)` - Hence, the dynamic multiplier for `\(j =1, \ldots , J\)` may be termed the IRF --- # Conclusion - Overview of fundamental concepts that we will apply throughout this course - South African data may contain trends and varying degrees of persistence - Provides challenge for regression models, as serial correlation in an error term provides inefficient standard errors - Considered statistical properties of many processes - Random walk: effect of errors do not disappear `\(\Rightarrow\)` difficult to forecast - `\(AR(p)\)` and `\(MA(q)\)`: effect of errors dissipate `\(\Rightarrow\)` over (extremely) long-term mean would represent a reasonable forecast and we just need to model the time taken to revert to mean - Other essential tools include difference equations, lag operators, autocorrelation functions, impact multipliers and impulse response functions