Lecturer Dr. Veronika Alhanaqtah ECONOMETRICS Lecturer Dr. Veronika Alhanaqtah

Topic 5. Time Series Time series and lags in economic models Stationary and non-stationary time series AR, MA, ARMA, ARIMA processes

1. Time series and lags in economic models When we analyze economic parameters we often use annual, quarterly, monthly and daily data. For example: annual GDP data annual net export data annual inflation rate monthly data on trading volumes daily data on production output In order to fulfil economic analyses rationally we need to systematize moments of receiving of corresponding statistical data. In this case we have to put in order our data in terms of time. Thus we get time series.

1. Time series and lags in economic models One-dimensional time series: we observe the dynamics of just one random variable in time. For example, only currency exchange rate or inflation rate Year GNP (mln JD) 2004 7320.8 2005 8285.1 2006 9163.9 2007 10966.6 2008 12539.6 2009 15885.5 2010 17178.8 2011 18609.6 2012 20288.8 2013 21689.6 2014 23611.2 2015 25141.2 Source: www.tradingeconomics.com

1. Time series and lags in economic models Multidimensional time series: we observe several variables in different moments of time, put in order. For example: Year Population (thousands) Unemployment rate (%) 2010 142 962 7.4 2011 142 914 6.5 2012 143 103 2013 143 395 5.5 Source: www.hse.ru

1. Time series and lags in economic models Definition: A time series is a sequence of data points, typically consisting of successive measurements made over a time interval.  Comment: Time series: random data plus trend, with best-fit line and different applied filters.

1. Time series and lags in economic models Mathematical notation We analyze a certain economic parameter Y. Its value in current moment of time t (period) is yt. Values of Y in succeeding (next) moments of time are yt+1, yt+2, …, yt+k, … Values of Y in previous (past) moments of time are yt-1, yt-2, …, yt-k, … When we study dynamics and relationship between such parameters, it is quite obvious that we can use as explainable variables (regressors) not only current values of variables, but also some past values, as well as time itself. We call such models as dynamic models. Variables, that are characterized by certain delay, we call lag variables.

1. Time series and lags in economic models  

1. Time series and lags in economic models In many cases affecting one factors by the others doesn’t happen immediately, but with some delay – time lag. Why lags do exist in economics: Psychological reasons which expressed in people’s behavior. Technological reasons. Institutional reasons. Mechanisms of formation of economic indicators.

2. Stationary and non-stationary time series One-dimensional time series: we observe the dynamics of just one random variable in time. Multidimensional time series: we observe several variables in different moments of time, put in order. Year GNP (mln JD) 2004 7320.8 2005 8285.1 2006 9163.9 2007 10966.6 2008 12539.6 2009 15885.5 2010 17178.8 2011 18609.6 2012 20288.8 2013 21689.6 2014 23611.2 2015 25141.2 Year Population (thousands) Unemployment rate (%) 2010 142 962 7.4 2011 142 914 6.5 2012 143 103 2013 143 395 5.5

2. Stationary and non-stationary time series Simulating of one-dimensional time series We observe a sequence of random variables (time series): y1, y2, y3, …, yt. In order to simulate we have to make assumptions. Without assumptions we can’t make conclusions. Basic assumption is about stationarity of series (its invariability in time). It doesn’t mean that y1=y2=…=yt. But it means that y1, y2, y3, …, yt have common characteristics in time.

2. Stationary and non-stationary time series  

2. Stationary and non-stationary time series  

2. Stationary and non-stationary time series Recall: Stationary process is a stochastic process the joint probability distribution of which does not change when shifted in time. Consequently, parameters such as the mean and variance also do not change over time and do not follow any trends. If a variable intersects its mean n times, we may say that the variable is stationary. Non-stationary process is a process which shifts in time.

2. Stationary and non-stationary time series Stationary process. “White noise” Example. Assume and are independent. A random vector is a white noise vector (white random vector) if its components have a probability distribution with the zero mean and constant finite variance, and are statistically independent. A necessary (but, in general, not sufficient) condition for statistical independence of two variables is that they are statistically uncorrelated (their covariance is zero):

2. Stationary and non-stationary time series Non-stationary processes: (1) Process with a deterministic trend (2) Process of random walk (drift) Why this series are non-stationary? The main problem is with dispersion (variance): The further this process goes in time, the less we are confident that it will stay in the vicinity of a zero point.

Non-stationary time series Process with a deterministic (stochastic) trend Process of random walk (drift) Mean is also not constant.

Random walk. Financial theory of random walk The basic idea of a random walk is that the value of the series tomorrow is its value today plus an unpredictable change. Because the path followed by Y(t) consists of random "steps" ε(t) that path is a "random" walk. The variance of a random walk increases over time, so the distribution of Y(t) changes over time. Because the variance of a random walk increases without bound, its population autocorrelations are not defined. However, the feature of a random walk is that its sample autocorrelations tend to be very close to 1. Random walk with drift is the extension of the random walk model to include a tendency to move (or "drift"), in one direction or the other. The financial theory of random walk is the theory, according to which stock prices, currencies and futures prices change randomly and cannot be predicted on the basis of past market data (the opposite to the technical analysis). This theory was proposed in the early 20th century and remained popular until the 1970s. Theory of random walk is a special case of the weak form of efficient markets hypothesis.

3. AR, MA, ARMA, ARIMA processes  

3. AR, MA, ARMA, ARIMA processes  

3. AR, MA, ARMA, ARIMA processes  

3. AR, MA, ARMA, ARIMA processes  

Economic time series Economic time series are recommended to analyze via computation their logarithms. 100∆ ln⁡Y(t) is the percentage change of a time series Y(t) between periods (t-1) and t. ∆ Y(t)=Y(t)-⁡Y(t-1) is the first difference of a series.

3. AR, MA, ARMA, ARIMA processes Comment: When two out of the three terms are equal to zero, the model may be referred to based on the non-zero parameter, dropping "AR", "I" or "MA" from the acronym describing the model. For example: ARIMA (1,0,0) is AR(1) ARIMA(0,1,0) is I(1) ARIMA(0,0,1) is MA(1) With the help of ARMA(p,q) we can compactly and precisely describe any stationary process. In ARMA-models coefficients are difficult to interpret but they may well be used for prediction.  

Time series Lecture summary Time series can be stationary and non-stationary. OLS is applied for stationary data; for non-stationary data OLS can’t be applied. Stationary time series are simulated by ARMA-models. Non-stationary time series (ARIMA) are reduced to stationary time series with the help of transformations (shift from absolute values to increments). It is reasonable to use AR, MA, ARMA processes when we already know the set of regressors and the functional form of a model but there is the problem of autocorrelation. In the regression analysis it is important to find long-term relationships. In case of autocorrelation we can also do this via the correction of standard errors (robust regression, i.e. model of correction of an equilibrium).