Econometrics. Week 11. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague

Transcription

1 Econometrics Week 11 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall / 30

2 Recommended Reading For the today Advanced Time Series Topics Selected topics from Chapter 18 In the next week Carrying out an Empirical Project Chapter 19 2 / 30

3 Today s Talk Selected advanced time series topics (we do not cover whole Chapter!) We could see that trends are important as they can lead to spurious regression. We need to have more knowledge to correctly estimate dependence in time series data. Most important result is that even if the two unit root series are independent, they can lead to statistically significant dependence when regressed one on another. Today, we will learn about concepts which will help us in dealing with this problem. We will see that even with statistical tests in hand, it may be difficult to make a decision. 3 / 30

4 Testing for Unit Roots The simplest approach to testing a unit root is to consider an AR(1) model: y t = α + ρy t 1 + ɛ t, for t = 1, 2,... where y 0 is initial value. {ɛ t } has zero mean, E(ɛ t y t 1, y t 2,..., y 0 ) Unit Root If {y t } follows AR(1) process, it has a unit root if and only if ρ = 1. If α = 0 and ρ = 1, {y t } follows a random walk without drift. If α 0 and ρ = 1, {y t } follows a random walk with drift, which means that E(y t ) is linear function of t. 4 / 30

5 Testing for Unit Roots cont. Although the processes with drift and without drift behave very differently, it is common to generally leave α unspecified. Testing for Unit Root In most of the cases, we consider testing the null hypothesis with one-sided alternative: H 0 : ρ = 1 H A : ρ < 1 In practice, 0 < ρ < 1 as ρ < 0 is unlikely in economic time series, and for ρ > 1, the process is explosive. When ρ < 1, {y t } is stable as Corr(y t, y t+h = ρ h 0) (BUT only for ρ < 1). 5 / 30

6 Testing for Unit Roots cont. Convenient way to test unit root is to subtract y t 1 from both sides of y t = α + ρy t 1 + ɛ t,: y t = α + θy t 1 + ɛ t, where θ = ρ 1. Then, we test H 0 : θ = 0, against H A : θ < 0. Unfortunately, a simple t-test is inappropriate, as y t 1 is I(1) under the null, but we can use appropriate critical values Dickey-Fuller distribution. This is a Dickey-Fuller (DF) Test for a unit root. Although theory used to obtain asymptotic critical values is too complicated at this level of studies, we can still use the tabulated values: For the significance level of 1%, 5% and 10%, it is -3.43, and resp. Thus, we reject the H 0 against H A at the 5% significance level, if tˆθ < / 30

7 Testing for Unit Roots cont. When we fail to reject the null hypothesis of unit root, we can only conclude that data do not provide strong evidence against unit root. We can also add dynamics by considering more lags: where γ 1 < 1. y t = α + θy t 1 + γ 1 y t 1 + ɛ t, Under the H 0 : θ = 0, {y t } follows stable AR(1) model. Under the alternative, H A : θ < 0, {y t } follows a stable AR(2) model. 7 / 30

8 Testing for Unit Roots cont. More generally, we can add up to p lags of y t to study full dynamics. It is extended version of Dickey-Fuller test, which we call Augmented Dickey-Fuller (ADF) test. ADF test y t = α + θy t 1 + γ 1 y t γ p y t p + ɛ t, with H 0 : θ = 0 and H A : θ < 0. This test is DF test, augmented with more lags. The critical values are the same as before. Logics is to include lags in order to clean up the serial correlation in y t. The problem is that if we include too many lags, we loose too many data! 8 / 30

9 Testing for Unit Roots with Trends If we have a series which has clear trend, we have to adjust for the trend first. There is a danger that we will mistake a trend stationary series for one with a unit root. Trend stationary process is I(0) but it has linear trend in mean. If we compute (A)DF test for the trending series with I(0), we will never reject the unit root, but series are not unit root. 9 / 30

10 Testing for Unit Roots with Trends Simple solution is to include the trend: y t = α + δt + θy t 1 + ɛ t, again, H 0 : θ = 0 and H A : θ < 0 If H 0 : δ = 0 is rejected, there a trend. We could also think of testing H 0 : θ = 0, δ = 0 jointly. Under the alternative, {y t } is a trend-stationary process. Because of trend, critical values for the DF test are wider. For the 1%, 5% and 10%, they are -3.96, and respectively. We can also augment this version of DF test for serial correlation. 10 / 30

11 Testing for Unit Roots with Trends There are several variants of unit root tests. We can omit intercept and set α = 0 for example. Usually, α 0 causes bias, but sometimes it is appropriate to omit it. We can also include other forms of trends, i.e. quadratic. Still, it is very difficult to conclude that the series come from the unit root process in the real-world data as you will learn. Most of the time we have to rely only on indicating a unit root, unless the unit root is really strongly rejected. 11 / 30

12 Spurious Regression As we have discussed earlier, it is common to find a spurious relationship between two trending time series. We can simply solve this problem by including the trend into the regression. But when we have I(1) series integrated of order 1, which are independent, regressing one on another will result in significant dependence. Consider two random walks: x t = x t 1 + a t and y t = y t 1 + ɛ t. Because a t and ɛ t are independent, also {x t } and {y t } are independent. But if regress ŷ t = ˆβ 0 + ˆβ 1 x t, ˆβ 1 will be significantly different from zero, even if there is no dependence between {x t } and {y t }. 12 / 30

13 Cointegration So it seems that regressing two I(1) variables may lead to spurious regression. The safe way is to first-difference the data and use the changes. But by the differencing we may loose some information. We can make regression involving I(1) variables potentially meaningful. Mathematics behind it is little more demanding, but we can explain basic intuition as it is very important concept. This finding was a major shift in modeling economic time series brought by two Nobelists (next slide). 13 / 30

14 Robert F. Engle and Clive W.J. Granger Robert F. Engle shared the Nobel prize (2003) for methods of analyzing economic time series with time-varying volatility (ARCH) with Clive W. J. Granger who recieved the prize for methods of analyzing economic time series with common trends (cointegration). Figure: (a) Robert F. Engle (b) Clive W.J. Granger 14 / 30

15 Cointegration cont. Cointegration If two processes {x t } and {y t } are I(1), then it is possible that there exist such a β that y t βx t is an I(0). In this case, we conclude that x and y are cointegrated with cointegration parameter β. For example, let s consider 2 time series which seems to have unit root. If we test their difference for unit root and find that they are I(0), the series are cointegrated. Thus if we know β, testing for cointegration is simple: We define s t = y t βx t and use DF test. If we reject a unit root, series are cointegrated. 15 / 30

16 Cointegration cont. But if β is unknown as in most of the applications, we have to first estimate it. After estimating β, we run a regression of û t on û t 1. and compare t-statistics on û t 1. If t statistics is below the critical value, we have evidence that y t βx t is an I(0) for some β. So y and x are cointegrated. Note If two series are cointegrated, spurious regression no longer arises and we can estimate their long-run relationship. If the two series are not cointegrated, no long-run relationship is present and regressing y t on x t results in spurious regression. We have to first difference in this case. 16 / 30

17 Cointegration cont. But what about the integrating coefficient β? y t = α + βx t + u t. As strict exogeneity assumption does not hold, we can not simply interpret the ˆβ OLS estimate. A simple solution is to put some exogenous variables into the regression: y t = α + βx t + φ 0 x t + φ 1 x t 1 + φ 2 x t 2 +γ 1 x t+1 + γ 2 x t+2 + ɛ t By construction, x t is now strictly exogenous and we can interpret β. This procedure works under the assumption of no trend in the tested time series. If there is trend present in the series, we need to add it into the initial regression that estimates β. In this case, the critical values will be different. 17 / 30

18 Cointegration cont. Several estimation procedures of cointegration were developed in the literature. The issues are more complicated and we will talk about them later in advanced courses. We will also not introduce the Error Correction Models or Engle-Granger two-step procedure. You have to wait. 18 / 30

19 Forecasting In lot of applications, we are not interested in estimating the casual or structural economic dependence. Our primary focus may be forecasting. First, we need to correctly specify the model and make sure estimation works. If we have inconsistent, or biased estimates, we can hardly use them for forecasting. 19 / 30

20 Forecasting cont. Let s consider one-day-ahead forecast. We would like to forecast y at time t + 1, y t+1. For this, we need to specify an information set observed at time t. In the simple AR(1) model, the information set will be the first lag, but we can specify and combine any other information set we like. So how do we know, which information set is the best? 20 / 30

21 Forecasting cont. We need to specify loss function based on the forecasting errors. For example for the one-day-ahead forecasts ŷ t+1, forecast error may be: ˆɛ t+1 = y t+1 ŷ t+1 We simply compare the real (observed) variable and its forecast. Simple approach may be to deal with errors symmetrically, and specify squared loss function, ˆɛ 2 t+1, or absolute errors ˆɛ t / 30

22 Forecasting cont. The goal is to minimize the loss function, conditional on the information set at time t: E [(y t+1 ŷ t+1 Information t )] If our forecast is expected value of y t+1 given available information at time t, loss function is minimal. We may easily generalize to multiple-step-ahead-forecast. In this case, one is interested in forecasting y at t + h period, y t+h. h is forecasting horizon. 22 / 30

23 Forecasting cont. So how do we forecast in the simple regression model: y t = β 0 + β 1 z t + u t. For obtaining the value of y t+1, we need to rewrite the equation: y t+1 = β 0 + β 1 z t+1 + u t+1. But this implies, that we have to explicitly know the future value of explanatory variable z t+1. In case we know the future value, i.e. in case of strong deterministic trend, forecasting is easy. In case we do not know the future value (as in most of the applications), we need to forecast it first. Thus equation takes form: E(y t+1 Information t ) = β 0 + β 1 E(z t+1 Information t ). 23 / 30

24 Forecasting cont. More intuitive is to include lagged values of explanatory variables in forecasting the y t+1 For example, we may specify the model as: y t = β 0 + α 1 y t 1 + γ 1 z t 1 + u t, where E(u t Information t 1 ). Until now, we have assumed only something we call point forecasts. Point forecasts are simply obtained by estimating the forecasting equation and then using the estimated parameters for forecasting. It is also possible to obtain forecast interval. 24 / 30

25 Forecasting cont. If we use lagged variables in the regression, model does not satisfy the CLM assumptions. Still, forecast interval is approximately normal, provided u t Information t 1 N(0, 1). Interval forecast The approximate 95% forecast interval is: ˆf n ± 1.96se(ˆɛ n+1 ), where ( [ se(ˆɛ n+1 ) = se( ˆf ] 2 1/2 n ) + ˆσ 2) where ˆf n is the forecast of y n+1, se( ˆf n ) is the standard error of the forecast and ˆσ 2 is standard error of the regression. 25 / 30

26 Comparing Forecasts In any forecasting exercise, we are not sure about several models. Should we include logs? levels? How many lags? etc... Lot of times, we specify several models and we need to compare them. We have to concentrate on the so-called out-of-sample testing. As you remember, first we fit the model on the in-sample data. We can look at criteria like R 2, but it only tells us if the model is correctly specified. Then, we take the estimated parameters and use them to forecast on the out-of-sample data. To correctly asses the forecasting model, we have to focus on the out-of-sample criteria. 26 / 30

27 Comparing Forecasts cont. Figure: British Pound returns data example in sample period out of sample period We use in-sample for estimation of the model Then, out-of-sample (data which are not included in the model) for forecasting evaluation. In fact, we need to save some data to be able to evaluate the forecasts. 27 / 30

28 Out-of-Sample Criteria Lot of times, model provides quite good fit, but not so good forecasts. Thus we need to use some criteria to evaluate the model forecasts. Root Mean Squared Error (RMSE) Let s consider we have n + m observations, where we use first n to estimate the model and save last m observations for forecasting. We can simply evaluate a loss function based on the forecast error ˆɛ n+h+1 : ( RMSE = m 1 ) m 1 h=0 ˆɛ2 n+h+1, 28 / 30

29 Out-of-Sample Criteria cont. RMSE is in fact the sample standard deviation of the forecasts errors. If we compare two models, the better one is the one with lower RMSE. BUT remember, always out-of-sample Mean Absolute Error (MAE) Second common measure is average of the absolute forecast errors: MAE = m 1 m 1 h=0 ˆɛ n+h+1 Again, model with smaller MAE is intuitively preferred one. 29 / 30

30 Thank you Thank you very much for your attention! 30 / 30