1 The linear regression model

The origin of modern time series analysis was pioneered by the contributions of Slutsky (1937), Yule (1926), and Yule (1927). These studies were essentially concerned with the interdependence of time series observations in a simple linear regression model:

\[\begin{eqnarray} y_{t} = \underbrace{x_t^{\top} \beta}_\text{explained} + \underbrace{\varepsilon_{t}}_\text{unexplained}\; , & \hspace{1cm} & t=1, \ldots , T \tag{1.1} \end{eqnarray}\]

In this case we are looking to explain the behaviour of a particular variable, \(y_t\), with the aid of a number of explanatory variables, \(x_t\), and parameters, \(\beta\). We also allow for certain behaviour that we are not going to be able to explain, which is represented by the error term, \(\varepsilon_t\). Note that by increasing the number of explanatory variables and potentially meaningless parameter values, it may be possible to reduce the error term, which would provide a larger value for the coefficient of determination. However, this practice would imply that we may provide an inadequate explanation of the behaviour of \(y_t\), which would be reflected by the out-of-sample forecasting performance of the model. This tradeoff between reducing the size of the errors that are associated with the estimation of the \(\beta\) parameters and the size of second-moment of \(\varepsilon_t\), is a central feature of this area of study.

One of the assumptions of the ordinary least squares estimate in such a model is that the errors (or shocks) should not be serially correlated, where \(\mathbb{E}[\varepsilon_t] = \mathbb{E}[\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2},\dots] = 0\) and \(\mathbb{E}\left[\varepsilon_i \varepsilon_{t-j}\right] = 0\), for \(j \ne 0\). If the errors are serially correlated then the least squares estimate of the coefficients are inefficient, which implies that the standard errors of the coefficient estimates are incorrect. Such a model would contain an essential mis-specification error, which should be corrected (possibly with the aid of a more accurate dynamic specification). Of course, one could employ certain GLS and FGLS estimates that have sought to contain the problems that arise when dealing with serially correlated errors; although, in most cases these procedures should be avoided.⁴

In most time series investigations, the variables that one would want to explain would exhibit some form of serial correlation (or persistence). For example, if economic output is large during the last quarter then there is a good chance that it is going to large during the current quarter. In addition, a change that arises in this quarter may only affect other variables at a distant point in time, while a particular shock during a specific period may affect the explanatory variable over several successive periods of time. In each of these cases, we would need to explicitly incorporate these features of the data in the specification of the regression model. If we fail to account for these features of the data in the part of the model that is to be explained, then the error term will usually incorporate some form of serial correlation.

During the course of this semester we will consider various ways of dealing with the essential features of time series data, to develop a more comprehensive understanding of the underlying dynamics that are may be present in many of these variables. Traditionally, a large part of time series analysis is concerned with forecasting and once we are able to describe the dynamics of the variable, the procedures that may be used to generate a forecast are usually relatively easy to apply. In addition, we can also make use of time series models to test various hypotheses (theories) that may be used to describe the past behaviour of a time series variable after accounting for the unique features that are present in the data.

2 The dynamic components of time series

An observed time series can be defined as a collection of random variables indexed according to the order they are realised over time. For example, we may consider a time series as a sequence of random variables, \(y_1 , y_2 , y_3 , \dots ,\) where the random variable \(y_1\) denotes the value taken by the series at the first time period, the variable \(y_2\) denotes the value for the second time period, and so on. In general, a collection of random variables that are indexed by \(t\) would be termed a stochastic process. For the purposes of our investigation, \(t\) will typically be discrete and vary over the integers, \(t = 0, 1, 2, \dots , T\) or some subset of these integers. Hence, the observed values of a stochastic process are termed realisations of this variable, which may take on many different functional forms.

Most of the data that we observe over time can be thought of as an aggregate representation of a number of different processes.⁵ As an example, consider the artificially generated data in Figure 1 that may represent monthly household expenditure. In the top panel, we have data for eight years of observed data, which is used to generate forecasts for the subsequent year. The result of a decomposition of a time series into the respective trend, seasonal and irregular components is depicted in the bottom panel. It incorporates a linear trend that is characterised by a upward sloping straight line, while the seasonal has a constant oscillating pattern that peaks in December. The irregular component is then used to display an element of random behaviour that may incorporate various shocks.

The equations that were used to artificially generate this data for periods 1 to 70, are as follows:

\[\begin{eqnarray} \nonumber \mathsf{Trend:} & \; & T_t = 1+0.05t \\ \nonumber \mathsf{Seasonal:} & \; & S_t = 1.6 \cos(t \pi 0.166) \\ \nonumber \mathsf{Irregular:} & \; & I_t = 0.5 I_{t-1} + \varepsilon_t \end{eqnarray}\]

where \(\varepsilon_t\) is a random disturbance, with zero expected mean and a predetermined variance.

The forecast that is show in the top panel relate to the sum of the forecasts that were derived from the individual components that are shown in the bottom panel. Note that when generating the forecast, it is assumed that the future value of the irregular will move towards zero, since the expected value of the error term, \(\varepsilon_t\), is equal to zero. Hence, if we are to assume that, \(\mathbb{E}_t[\varepsilon_{t+1}]=0\), it will be the case that \(I_{t+\infty} \rightarrow 0\).

These equations may be regarded as the equations of motion, since they describe the behaviour of a system throughout time. It has been suggested that by decomposing a time series into separate components (or processes) we are often able to provide a better description of the aggregate time series, which would usually provide more accurate forecasts.⁶ In addition, after identifying each of the dynamic components in the time series variable, we may also be able to generate robust parameter estimates despite any serial correlation in the data.

3 Examples of time series variables

Time series variables may be measured over a wide range of frequencies. For example, we could encounter hourly, daily, weekly, monthly, quarterly or yearly data. The frequency of the data has important implications for both the type of analysis that could be performed and the type of transformation that should be performed (where necessary). For example, when seeking to conduct an analysis on the business cycle, monthly or quarterly data is preferred. In such a case, the use of annual data may not provide a useful insight into the properties of the data. However, when seeking to understand the behaviour of stock market returns, daily or even hourly data may be of importance. In other instances, when analysing long-run growth, the use of yearly data may suffice as the defining features of growth evolve slowly over time.

Although it is relatively simple to aggregate higher frequency data to lower frequency (e.g. from monthly to quarterly), it is more difficult to obtain reasonable estimates of higher frequency data from data that was originally captured at a lower frequency (i.e. where methods of interpolation are used).⁷ Thus, the frequency of available data will usually influence the type of economic or financial analysis that one is able to perform.

In addition, a further important feature of time series data is that the sample may vary. For example, consumer price data for South Africa only goes back to 1975, while measures of gross domestic product (GDP) that are published by the South African Reserve Bank (SARB) start in 1960.⁸ In addition, the estimates of GDP are usually reported up to two quarters in areas, while we are able to obtain a measure of consumer prices from Statistics South Africa (StatsSA) for the preceding month. As many results in statistics and econometrics depend on the use of a sufficient sample size, this may be of importance in certain cases.

When using time series data, it is extremely important to always consider the potential information content that may be contained in the data. For example, most economic data contains trends, seasonals, and irregular components (when measured in levels). This data is usually measured in discrete time (with relatively long intervals) and may be subject to revision. The variables may also be expressed as rates, indices or totals and before they may be used in an analysis, it may be necessary to transform the variable by calculating the growth rate \([\log(GDP_{t}/GDP_{t-1})]\), or the annualized rate \([(1+(i_t/100))^{(1/12)}-1]\). Alternatively, it may be necessary to remove the stochastic trend or derive a measure for a cyclical component.

Data on financial instruments and indices can be overwhelming and there are a number of considerations that need to be taken into account.⁹ As such, you may wish to answer the following questions before you start to construct a model for the data: Does the data contain true trading prices, quotes, or proxies for trading prices? Are we only be interested in buyer initiated (ask) or seller initiated (bid) orders? Do the prices include transaction costs, commissions and the effects of tax transfers? Is the market sufficiently liquid? Have the prices been adjusted for inflation, or have they been correctly discounted? At what frequency do you want to measure trading activity/returns?

Note also that high-frequency data for the price of a security that is seldom traded (i.e. when the security is relatively illiquid) is not going to be of much use. In addition, if we were only to use three years of high frequency data over the period of the global financial crisis, then such a sample would be dominated by the irregular trends that existed over this period of time. Of course, such a model would not be of much use when looking to derive an inference that relates to the normal trading behaviour that may have existed before or after the crisis.

Figure 2 depicts a number of economic time series variables that describe various aspects of the South African economy. In this case, we can see how some of these variables are dominated by the trend, as is the case for GDP, while other variables display differing degrees of volatile around a mean that is close to zero. Others, such as non-seasonally adjusted consumption clearly incorporate strong seasonal behaviour. Accounting for these features of the data in an appropriate manner is one of the primary concerns of those involved in the field of time series research.

4 Prominent processes in time series data

Since a time series may be thought of as a combination of different processes, it is worth spending some time discussing the functional forms of commonly encountered processes.¹⁰

In terms of a formal definition, a time series may be regarded as a collection of observations indexed by the date of each realisation. For example, where the starting point in time is \(t = 1\) and the ending point is \(t =T\), we would have the time series process,

\[\begin{eqnarray} \nonumber \left\{ y_{1}, y_{2}, y_{3}, \ldots , y_{T}\right\}. \end{eqnarray}\]

The time index in the above expression could be of any frequency (e.g. daily, quarterly, etc.). When working with simulated time series processes, this (finite) sample may be regarded as a subset of an infinite sample, indexed by \(\left\{ y_{t}\right\}_{t=-\infty }^{\infty}\). A particular observation within any time series process is then identified by referring to the \(t\)-th element.

4.2 Stochastic processes

With a stochastic process there is certain degree of indeterminacy that relates to the expected future values of a particular process. This indeterminacy may be described by a probability distribution. Hence, even in the case where the initial condition is known, there are many possible values that could be realised by the process, where some paths may be more probable than others.

If the sequence \(y_{t}\) is termed a random time series variable, it is inferred that we are dealing with a sequence of random variables that are ordered in time. We may call such a sequence a stochastic process. Examples of stochastic processes include: white noise processes, random walks, Brownian motions, Markov chains, martingale difference sequences, etc.

4.2.1 White noise process

The building block for most time series models is the white noise process, which is often used to describe the distribution of the errors in the model. It is defined as a sequence of serially uncorrelated random variables with zero mean and finite variance. One may choose from a number of different distributions to describe the scattering of such a process. When the errors are assumed to follow a normal distribution, the process is often termed a Gaussian white noise process. A slightly stronger condition is that the individual realisations are independent from one another. Such an independent white noise process may be denoted \(\varepsilon_t\), where,

\[\begin{eqnarray} \nonumber \varepsilon_t \sim \mathsf{i.i.d.} & \mathcal{N}(0, \sigma_{\varepsilon_t}^2) \end{eqnarray}\]

This assumption leads to three important implications:

1. \(\mathbb{E}[\varepsilon_t] = \mathbb{E}[\varepsilon_t | \varepsilon_{t-1}, \varepsilon_{t-2}, \dots ] =0\)
2. \(\mathbb{E}[\varepsilon_t \varepsilon_{t-j}] = \mathsf{cov}[\varepsilon_{t} \varepsilon_{t-j}] = 0\)
3. \(\mathsf{var}[\varepsilon_{t}] = \mathsf{cov}[\varepsilon_{t} \varepsilon_{t}] = \sigma_{\varepsilon_t}^2\)

where, the first and second properties refer to the absence of predictability and auto-correlation. The third property refers to conditional homoskedasticity or a constant variance, where the subscript is used to infer that the variance is specific to the process \(\varepsilon_t\). It is not meant to infer that it varies over time as \(\varepsilon_t\) varies over time. Note that if \(\varepsilon_t\) is unusually high, there is no tendency for \(\varepsilon_{t+1}\) to be unusually high or low. Hence, this process doesn’t exhibit any form of persistence.

An example of time series with zero mean and constant variance is plotted in Figure 3 for 100 observations. To construct this series, we simply simulated a certain set of values for \(\left\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3,} \dots , \varepsilon_{T}\right\}\) with a sequence of random numbers that are distributed \(\mathcal{N} (0, 1)\).

Figure 3: Gaussian White Noise Process

When comparing the white noise process in Figure 3 to the real South African data in Figure 2, we note that there are some significant differences. In particular, the white noise process does not seem to share the persistence of the interest rate series, where there are large and prolong departures from the mean in the data. In this sense, the white noise process would appear to be just noise.

Of course, one may make use of a number of other probability distributions, such as a students-\(t\), Cauchy, Poisson, etc. In addition, we may also wish to consider the effects of relaxing the assumption of constant variance in certain instances.

4.2.2 Random walk

The term random walk stems from the fact that the value of the time series at time \(t\) is the value of the series at time \(t - 1\), plus a completely random movement determined by \(\varepsilon_t\). Such a process may be described as,

\[\begin{eqnarray} y_t = y_{t-1} + \varepsilon_t \tag{4.1} \end{eqnarray}\]

Figure 4: Random Walk - Simulated Time Series

This process may be simulated repeatedly and on each occasion you would obtain significantly different results.¹¹ An example of such a series if depicted in Figure 4. Note that where the first value of process is very small, \(y_{0} \sim 0\), then we may use repeated substitution to rewrite (4.1) as a cumulative sum of white noise variates (which is the solution to the above expression),

\[\begin{eqnarray} \nonumber y_t = \sum^t_{j=1} \varepsilon_j \end{eqnarray}\]

Figure 5 shows the effect of a shock in the initial period, \(\varepsilon_0 = 1\). In this case, if we assume that the starting value for \(y_{-1} = 0\), and there are no subsequent shocks after this period, then we note that the shock will have a permanent effect on \(y_t\).

Figure 5: Random Walk - Effect of Shock [\(y_{-1}=0, \varepsilon_0 = 1\) & \(\{\varepsilon_1, \dots\} = 0\)]

4.2.3 Random walk with drift

The random walk plus drift incorporates the additional constant \(\mu\), which would give the appearance of a time trending variable,

\[\begin{eqnarray} y_t = \mu + y_{t-1} + \varepsilon_t \tag{4.2} \end{eqnarray}\]

In Figure 6 we have include different values for \(\mu\). The dotted line, where \(\mu = 1.2\) is much steeper than the solid line where \(\mu = 0.5\). In this case, when the starting value is small, we may use repeated substitution to express this variable as a cumulative sum of white noise variates, plus a time trend;

\[\begin{eqnarray} \nonumber y_t = \mu \cdot t + \sum^t_{j=1} \varepsilon_j \tag{4.3} \end{eqnarray}\]

Figure 6: Random Walk plus Drift - Simulated Time Series [Solid: \(\mu=0.5\) & Dotted: \(\mu = 1.2\)]

The effect of the shock on the random walk plus drift is depicted in Figure 7. Once again, we note that the effect of a shock would be permanent, but with the addition of a deterministic time trend. Hence, in this case the solution would suggest that the effect of a positive shock during the initial period would result in a continued increase over time.

Figure 7: Random Walk with Drift - Effect of Shock [\(y_{-1}=0, \mu = 1.2, \varepsilon_0 = 1\) & \((\varepsilon_1, \dots) = 0\) ]

Figure 8 depicts examples of three random walks. To generate each series we have simulated different white noise disturbances that are represented by \(\varepsilon_{t}\), while iterating the process in (4.3) from the starting value of \(y_{0}=0\) to \(y_T\). At each point in time, a new random draw of the disturbances is added to the process.

Figure 8: Different Random Walks

As can be seen in the graph, all three processes start off from values that are close to zero, but then wander off in different directions. For this reason, random walks are typically hard to predict or analyse. However, we cannot simply disregard these processes as most macroeconomic variables may exhibit this type of nonstationary in levels, which may give rise to random walk behaviour. How we choose to deal with this characteristic of the data will be considered on several occasions during the remainder of this semester.

4.2.4 Autoregressive process

When the present value of a time series process is a linear function of the immediate past, we would make use of an AR(1) representation, where the numeral refers to the fact that we are only going to include a single lag of the dependent variable on the right-hand side. i.e.,

\[\begin{eqnarray} y_{t}= \phi y_{t-1} + \varepsilon_{t}, \hspace{1cm} \varepsilon_t \sim \mathsf{i.i.d.} \mathcal{N}(0, \sigma_{\varepsilon_t}^2) \tag{4.4} \end{eqnarray}\]

In this case, such a representation would suggest that we know something precise about the conditional distribution of \(y_t\) given \(y_{t-1}\).¹² Once again, when the first value of process is very small, repeated substitution would allow us to rewrite (4.1) as follows,

\[\begin{eqnarray} \nonumber y_t = \phi^j \sum^t_{j=1} \varepsilon_j \end{eqnarray}\]

Figure 9: AR(1) - Simulated Time Series [\(\phi = 0.9\)]

Note that when the coefficient is less than unitary, \(\phi<|1|\), then this particular time series will converge on the mean value, which is zero in this case. Of course, if we included a mean value of \(\mu\), such that the autoregressive process is written as, \(y_{t}= \mu + \phi y_{t-1} + \varepsilon_{t}\), and the process would converge on the value of a constant that is different to zero. An example of an autoregressive process is depicted in Figure 9, where we note that the deviations from the mean are slightly more pronounced than in the case of the white noise process.

Figure 10: AR(1) - Effect of Shock [\(\phi = 0.9\)]

Figure 10 is then used to show the effect of a shock to the autoregressive process that is described in (4.4). Note that the speed of convergence on the mean value of zero is largely dependent upon the size of the coefficient, \(\phi\). In addition, if you were to compare the random walk plus drift with the autoregressive process with a constant, you would note that the role of \(\mu\) is remarkably different.

Of course, one could include several lags in an autoregressive model, where it could be take the form of an AR(\(p\)) structure.

4.2.5 Moving average process

The moving average process is often used to describe a time series by the weighted sum of the current and previous white noise errors. These models are useful when seeking to describe processes where it takes a bit of time for the error (or shock) to dissipate. The MA(1) model may be represented by the expression,

\[\begin{eqnarray} \nonumber y_t = \varepsilon_t + \theta \varepsilon_{t-1}. \end{eqnarray}\]

Figure 11: MA(1) - Simulated Time Series [\(\theta = 0.7\)]

Figure 12: MA(1) - Effect of Shock [\(\theta = 0.7\)]

This model may be used to describe a wide variety of stationary time series processes, and may incorporate several lags, where it is expressed in the more general form of MA(\(q\)).¹³

4.2.6 ARMA processes

Autoregressive and Moving Average processes may be combined to form what is termed an ARMA(\(p\),\(q\)) process. For example, we may express an ARMA(1,1) process as,

\[\begin{eqnarray} \nonumber y_t = \phi y_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1}. \end{eqnarray}\]

where an ARMA(\(p\),\(q\)) would take the form,

\[\begin{eqnarray} \nonumber y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \theta_2 \varepsilon_{t-2} + \dots + \theta_q \varepsilon_{t-q}. \end{eqnarray}\]

This framework has been popularized by Box and Jenkins (1979), who developed a methodology for the identification of observed real world data that may be described by such a model. Note that the AR(\(p\)), MA(\(q\)) and ARMA(\(p\),\(q\)) may characterise some of the South Africa data that was depicted in Figure 2. Therefore, this framework will receive quite a bit of coverage over the next few weeks, as it will serve as a point from which we can consider various extensions.

4.2.7 Long memory and fractional differencing

The conventional AR(\(p\)), MA(\(q\)) and ARMA(\(p\),\(q\)) processes are often termed short-memory processes as the coefficients in the representation are dominated by exponential decay. The properties of long memory (or persistent) time series processes has been considered in Hosking (1981) and Granger and Joyeux (1980). These processes are regarded as an intermediate compromises between the short memory ARMA(\(p\),\(q\)) type models and the fully integrated nonstationary processes (such as random walks). These processes are usually used to describe variables where there are usually long periods when observations tend to be at a reasonably high level and similar long periods when observations tend to be at a low level.

For such a model it is assumed that the sample correlation decay at a rate that is approximately proportional to \(k^{-\lambda}\) for some \(0 < \lambda < 1\). This is noticeable slower than the rate of decay that pertains to realisations of an AR(1) process. The closer \(\lambda \rightarrow 0\), the more pronounced is the long memory.

5 Conditional and unconditional moments

When working with stochastic time series processes we are usually concerned with the distribution or density of these variables. These distributions may be summarised by the first-order (mean) and second-order (variance) moments; although higher order moments may also be of interest (i.e. skewness and kurtosis). When considering the density of these variables, an important distinction should be made between their unconditional and conditional distribution. In what follows, we make use of some simple definitions and notations for these features of a time series variable.

6 Stationarity, autocorrelation functions & Q-statistics

Stationarity is a fundamental concept in time series analysis, and the distinction between the properties of a nonstationary and stationary time series is of great importance. In addition, it is also important to define the degree of stationarity as there are various different forms. For this purpose, a time series is termed strictly stationary if for any values of \(\{j_{1}, j_{2}, \dots , j_{n}\}\), the joint distribution of \(\{ y_{t}, y_{t+j,1} , y_{t+j ,2}, \dots , y_{t+j,n} \}\), depends only on the intervals separating the dates \(\{ j_{1}, j_{2}, \dots , j_{n}\}\), and not on the date itself, \(t\). 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 \(j\),

\[\begin{eqnarray} \nonumber \mathbb{E}\left[ y_{t}\right] &=&\bar{y} \\ \nonumber \mathbb{E}\left[ \left( y_{t}-\bar{y} \right) \left( y_{t-j}-\bar{y} \right) \right] &=&\mathsf{cov}(y_{t},y_{t-j}) \end{eqnarray}\]

In what follows, when we refer to a stationary process, we will make use of the less limiting conditions of covariance stationarity.

One of the essential features of a nonstationary time series is that the mean is not required to be constant. For example, the process \(y_{t}=\alpha t+\varepsilon_{t}\) would not be stationary, as the mean of the process clearly depends on \({t}\). To provide a formal presentation of this result, we could compute the unconditional mean of the process,

\[\begin{eqnarray} \nonumber \mathbb{E}\left[ y_{t}\right] =\mathbb{E}[\alpha t+\varepsilon_{t}]=\mathbb{E}[\alpha t]+\mathbb{E}[\varepsilon_{t}]=\alpha t \end{eqnarray}\]

which confirms that the mean of the process depends on \(t\).

To provide a second example, consider the difference equation in (5.1), which takes the form \(y_{t}=\phi y_{t-1}+\varepsilon_{t}\). If this process is stationary then it require that \(|\phi |<1\). In such cases, when calculating the unconditional summary statistics that are provided in equations (5.3), neither the mean nor the covariances depend on time. In other words, this process would be stationary, as there are no time subscripts in the expressions for the first- and second-order moments of the unconditional distribution (and similarly so for the covariance).

6.3 Q-statistic

The Box-Ljung \(Q\)-statistic is often used as a test of whether the series is white noise. Formally, it tests whether any of a group of autocorrelations of a time series are different from zero. Instead of testing randomness at each distinct lag, it tests the “overall” randomness based on a number of lags. It may be expressed as,

\[\begin{eqnarray} \nonumber Q(k) = T(T+2) \sum_{j=1}^{k} \frac{\rho_j^2}{T-j} \end{eqnarray}\]

where \(\rho\) refers to the residual autocorrelation from lag \(j\). In this case we would express \(\rho\) as,

\[\begin{eqnarray} \nonumber \rho_k = \frac{\sum_{t=1}^{T-k} (\varepsilon_t - \bar{\varepsilon})(\varepsilon_{t+k} - \bar{\varepsilon} )}{\sum_{t=1}^{T} (\varepsilon_t - \bar{\varepsilon})^2} \end{eqnarray}\]

where \(\bar{\varepsilon}\) is the mean of the \(T\) residuals.

Figure 13: Different Random Walks

This statistic is tested against a \(\chi^2\) distribution with \((k-w+1)\) degrees of freedom (where \(w\) is the number of terms in an ARMA model). If \(Q > \chi^2_{k-w+1}\) then we can reject the null hypothesis that the data is random white noise. Note that when we use this statistic in an ARMA setup, then we consider whether the residuals are white noise (and as such the test is performed on the residuals and not the series itself).

The results of this test, for various processes, are contained in Figure 13.

7 Impact multipliers and impulse responses

In many macroeconomic and financial studies, we often want to investigate the causes and effects of particular events. For example, we may want to estimate the response of the exchange rate to an unexpected \(1\)% increase in the short-term interest rate.

Under the assumption of stationarity, we can show that any autoregressive process can be written as an infinite order moving average equation.¹⁵ This would imply that an AR(1) process may be written as,

\[\begin{equation}\nonumber y_{t}=\varepsilon_{t}+\phi \varepsilon_{t-1}+\phi ^{2}\varepsilon_{t-2}+ \dots = \overset{\infty }{\underset{j=0}{\sum }}\phi^{j}\varepsilon_{t-j} \end{equation}\]

which would suggest that \(y_{t}\) can be described by observed realisations of the errors (or shocks) during past and present periods of time.

If we assume that the dynamic simulation started at time \(j\), taking \(y_{t-(j+1)}\) as given, the effect of a change in the initial shock (assuming the rest remains the same) on \(y_{t}\) is then,

\[\begin{equation} \frac{\partial y_{t}}{\partial \varepsilon_{t-j}}=\phi ^{j} \tag{7.1} \end{equation}\]

This expression is termed the dynamic multiplier. As seen from (7.1), the dynamic multiplier only depends on \(j\), \(\varepsilon_{t-j}\), and the current value of the variable, \(y_{t}\). Importantly, this expression does not depend on \(t\), the date of any observation.

8 Conclusion

This chapter provides a brief overview of a few fundamental concepts that we will apply throughout this course. After looking at a few examples of observe real world data, we noted that many economic and financial variables contain trends, seasonals and various different types of persistence. This form of serial correlation may present a number of challenges when using a simple regression model, as the the existence of serial correlation in the error term provide inefficient standard errors for the coefficient estimates.

We also considered the statistical properties of a number of processes. In the case of a random walk, the effect of errors (or shocks) do not disappear with time. As a result, it would be difficult to generate a reasonable forecast for these processes. In contrast, the effect of errors (or shocks) in autoregressive and moving average processes dissipate with time.

The latter sections of this chapter included a brief introductory discussion that relates to some of the important tools that will be used throughout this course. These include autocorrelation functions, impact multipliers and impulse response functions.

9 References