Econometrics for Policy Analysis A Train The Trainer Workshop Oct 22-28, 2016

Econometrics for Policy Analysis A Train The Trainer Workshop Delivered by Dr. Nathaniel E. Urama Department of Economics, University of Nigeria, Nsukka

Loading Time Series data in E-views: Review For the purpose of this training, we will make use of E-views. The first thing therefore is to load the time series data in E-views. Steps: Save the data in Excel (xlsx or csv comma delimited). Open the E-view software; For Version 8 and above, click on Open Foreign files, select the file, open, next, next, finish. Double click any of the series to cross check. If you see NA in place of data values, your loading was wrong, if you see the data values, you are good to go!! Delivered by Dr. Nathaniel E. Urama 2

Loading Time Series data in E-views: Review cont. For Versions below 8, Click on file, new, workfile On the Workfile create Dialogue box below, select the frequency, Enter start date e.g.(1970) oor 1970Q1 and the end date, type the workfile name e.g. (TTT1), ok. Delivered by Dr. Nathaniel E. Urama 3

Loading Time Series data in E-views: Review cont. On the workfile TTT1 user interface, click; Proc, Import, Read text lotus excel. Select the file containing the data saved in csv comma delimited, Enter the number of series, ok. Delivered by Dr. Nathaniel E. Urama 4

Class Activity 1 1. Load the data set of Nigerian Macro Variables provided. 2. Follow the above steps to load the data set in EViews depending your version. 3. Revision on how to: i. manipulate the data like; logging, differencing and lagging a variable; generating another variable that is a mathematical combination of other variables etc. ii. Plotting graph/ Chats of the Series to visualise there statistical properties. iii. Test the variables for Structural breaks iv. Test the variables for Stationarity or level of Integration. v. Test for stationarity Under structural Breaks. Delivered by Dr. Nathaniel E. Urama vi. Cointegration tests 5

Univariate Time Series models AR, MA, ARMA, ARIMA, ARMAX, ARIMAX or ARFIMA? What are these Models? Compare to structural Models a-theoretical Why use them? Serial correlation: presence of correlation between residuals and their lagged values OLS is no longer efficient among linear estimators. Standard errors computed using the textbook OLS formula are not correct, and are generally understated. If there are lagged dependent variables on the right-hand side of the equation specification, OLS estimates are biased and inconsistent. Lack of data date for structural models ( absence, incorrect frequency, immeasurable) Better for Forecasting Delivered by Dr. Nathaniel E. Urama 6

AR(p) Model: A review An autoregressive model is one where the current value of a variable depends upon only its previous values and a white noise error term. AR p Y t = α 1 Y t 1 + α 2 Y t 2 + + α p Y t p + ε t p = j=1 α j Y t j + ε t (1) Using a lag operator L such that L k Y t = Y t k p The AR(p) is given as: Y t = j=1 α j L j Y t + ε t p Y t j=1 p The term: 1 j=1 the AR model. α j L j p Y t = ε t or 1 j=1 α j L j Y t = ε t (2) α j L j is the characteristics polynomial of AR 1 is given as 1 αl Y t = ε t (3) Delivered by Dr. Nathaniel E. Urama 7

MA(p) Model : A review An MA(q) model a linear combination of white noise processes, so that yt depends on the current and previous values of a white noise disturbance term. MA q Y t = ε t + θ 1 ε t 1 + θ 2 ε t 2 + + θ p ε t q q = ε t + j=1 = θ(l)ε t θ j ε t j (4) q For θ L = 1 + j=1 θ j L j (5) q The term: 1 + j=1 θ j L j is the characteristics polynomial of the MA model. 8 where ε t are the independent and identically distributed innovations for the process Delivered by Dr. Nathaniel E. Urama

MA(p) Model : A review The distinguishing properties of the moving average process of order q given above are: 1. E y t = μ 2. var y t = γ 0 = 1 + θ 1 2 + θ 2 2 + + θ q 2 σ 2 3. cov y t = γ s = 1 + θ 2 1 + θ 2 2 + + θ 2 q σ 2 for s = 1,2,, q 0 for s > q 9 Delivered by Dr. Nathaniel E. Urama

ARMA model: A review ARMA p, q is a combination of Ar(p) amd MA(q) as follows: p ARMA p, q Y t = j=1 q α j Y t j + ε t + j=1 θ j ε t j (6) ARMA(1,1) is given as: 1 αl Y t = 1 + θl ε t (7) Seasonal AR and MA Terms: Due to seasonal patterns in most monthly and quarterly data, Box and Jenkins (1976) recommend the use of seasonal autoregressive (SAR) and seasonal moving average (SMA) terms in the ARMA process. SAR p is a seasonal AR term with lag p and it adds to an existing AR, a polynomial with lag p given as 1 p L p : A second order AR process for quarterly data can be written as; 1 α 1 L 1 α 2 L 2 1 4 L 4 Delivered by Dr. Nathaniel E. Urama Y t = ε t (8) 10

AR, MA and ARMA, : A review (8) on expansion will give: Y t = α 1 Y t 1 + α 2 Y t 2 4 Y t 4 α 1 4 Y t 5 α 2 4 Y t 6 + ε t (9) For seasonal moving average with lag q, the resulting MA lag structure is obtained from the product of the lag polynomial specified by the MA terms and the one specified by any SMA terms. For a second order MA without seasonality, the process is written as : Y t = ε t + θ 1 ε t 1 + θ 2 ε t 2 2 = ε t + j=1 θ j ε t j (10) This in the lag form is given as: Y t = 1 + θ 1 L 1 θ 2 L 2 ε t (11) Delivered by Dr. Nathaniel E. Urama 11

AR, MA and ARMA, : A review If the data for (11) is quarterly for example, we introduce the SMA(4) given as 1 + φ 4 L 4 in the MA term. This will give: Y t = 1 + θ 1 L 1 θ 2 L 2 1 + φ 4 L 4 ε t (12) (12) on Expansion will give: Y t = ε t + θ 1 ε t 1 + θ 2 ε t 2 + φ 4 ε t 4 + θ 1 φ 4 ε t 5 + θ 2 φ 4 ε t 6 (13) The parameter φ is associated with the seasonal part of the MA process. Delivered by Dr. Nathaniel E. Urama 12

ARIMA and ARIMAX models The AR, MA and ARMA models discussed before assumes that the series in question is at least weakly stationary. Why? (see Gujarati, 2004, pp. 840) Since most time series are not stationary, there is need to account for this in our ARMA model. Hence, the need for ARIMA model. In our previous class, a series that must be differenced d times for it to become stationary is said to integrated of order d i.e. I(d) ARIMA (p,d,q) is an ARMA(p,q) model of non-stationary series differenced d times to make it stationary. Delivered by Dr. Nathaniel E. Urama 13

Estimating ARIMA models:the BJ [Box Jenkins] Methods Revisited How do we identify the value of P, d and q for an ARIMA(p, d, q) models? The BJ methodology has an answer and consists of the following steps: Differencing to achieve Stationarity Identification Estimation Diagnostic Checking Forecasting Delivered by Dr. Nathaniel E. Urama 14

The BJ [Box Jenkins] Methods Differencing the Series to achieve stationarity Identify the model to be tentatively estimated Estimate the parameters of the tentative model Diagnostic Checking of the adequacy of the model ------Is the model adequate? No Yes Use the model for Forecasting and control Delivered by Dr. Nathaniel E. Urama 15

The BJ [Box Jenkins] Methods cont. 1. Differencing to achieve stationarity: If a series is integrated of order d, it has to be differenced d times to make it stationary. 1. Identification: This involves determining the order of the model required to capture the dynamic features of the data. 2. The chief tools are autocorrelations (AC), the and partial autocorrelations (PAC), and the correlogram of the series. Covariance at lag K AC at lag k is given as ρ k = With γ k = t=1 n k Y t Y n Y t+k Y, variance = γ k γ 0 γ 0 = Y t Y 2 n A plot of ρk against k is known as the sample correlogram. Note if n is small, use n-1 as the divisor. Asymptotically however, both give same result 16 Delivered by Dr. Nathaniel E. Urama

Finding the value of d in ARIMA (p, d, q) Model The autocorrelation is the correlation coefficient of the current value of the series with the series lagged a certain number of periods. For the purely white noise(stationary) process the autocorrelations bars/values are not significantly different from zero at various lags. For pure RWM, the autocorrelation coefficients at various lags are statistically different from zero. Generating AC and PAC in Eviews: Click Quick, series statistics, correlogram, on the dialogue box that will appear, select level, ok Or Double click the series, View, correlogram, Level, Enter the lag length OK. Delivered by Dr. Nathaniel E. Urama 17

Correlogram of a white noise random process Delivered by Dr. Nathaniel E. Urama 18

Correlogram of a random walk time series Delivered by Dr. Nathaniel E. Urama 19

Finding the value of d in ARIMA (p, d, q) Model cont. If a time series exhibits white noise, the sample autocorrelation coefficients ρ k are approximately ρ k ~N 0, 1 n Hence, following the properties of the standard normal distribution, the 95% confidence interval for any (population) ρ k is: ρ k ± 1.96 1 n If the preceding interval includes the value of zero, we do not reject the hypothesis that the true ρk is zero. If this interval does not include 0, we reject the hypothesis that the true ρk is zero. Alternatively, we can use the asymptotic confidence limit for the ρ k which is ± 0.2089. 20 Delivered by Dr. Nathaniel E. Urama

Finding the value of d in ARIMA (p, d, q) Model cont. If the autocorrelation coefficients ρ k are statistically zero, the value of d=0. Otherwise, it may not be zero. Run the correlogram for the 1 st, 2 nd, until the dth higher order difference of the series where autocorrelation coefficients ρ k becomes statistically zero. Steps in Eviews: Double click the series, View, correlogram, select the order and enter the lag length, ok For series needing more than 2 nd differencing, Quick, series statistics, correlogram, type D(Y, n) as the series name where y is the series name and d is the number of differencing, select level, enter the lag length, ok. 21 Delivered by Dr. Nathaniel E. Urama

Finding the value of d in ARIMA (p, d, q) Model cont. For series having seasonal terms, click Quick, series statistics, correlogram type d(y, n, s) as the series name where y is the series, with seasonal difference at lag s, and d the number of differencing, level, enter the lag length, ok. d y, n, s = d 1 L 2 (1 L s )y More accurately, conduct the structural break and the stationarity test to determine the order of the integration of the series which is the d. Why Structural Break? The presence of structural breaks in a series may lead to wrong conclusion that the series is not stationary. See the example below for the annual series of OER in our data set. Delivered by Dr. Nathaniel E. Urama 22

The BJ [Box Jenkins] Methods cont. Correlogram of OER below show that it is not stationary Delivered by Dr. Nathaniel E. Urama 23

The BJ [Box Jenkins] Methods cont. Correlogram of OER below show that it is stationary after the first difference i.e., the D may be 1 Delivered by Dr. Nathaniel E. Urama 24

The BJ [Box Jenkins] Methods cont. The ADF unit root test confirm that ORE is I(1) Delivered by Dr. Nathaniel E. Urama 25

Finding the value of d in ARIMA (p, d, q) Model cont. The ADF results are confirmed by the KPSS results below. Delivered by Dr. Nathaniel E. Urama 26

Finding the value of d in ARIMA (p, d, q) Model cont. A test of the series OER for structural breaks shows the rejection of null of no structural break as shown in the next 2 Slides below. Delivered by Dr. Nathaniel E. Urama 27

Finding the value of d in ARIMA (p, d, q) Model cont. A test of the series OER for structural Using the multiple breakpoint test. Delivered by Dr. Nathaniel E. Urama 28

Finding the value of d in ARIMA (p, d, q) Model cont. A Look at the series graph reveals the breaks. Delivered by Dr. Nathaniel E. Urama 29

Finding the value of d in ARIMA (p, d, q) Model cont. Zivo-Andrews unit root test shows a rejection of the null hypothesis that OER has a unit root with a structural break. This shows that OER may be I(0). Delivered by Dr. Nathaniel E. Urama 30

Finding the value of d in ARIMA (p, d, q) Model cont. In E-views 9 and above, one can conduct unit root with structural break test directly. The result as shown below also show that considering the structural breaks, OER may be I(0). Delivered by Dr. Nathaniel E. Urama 31

Finding the values of p and q in ARIMA (p, d, q) Model The PAC measures the correlation of Y values that are K periods apart after removing the correlation from the intermediate lags. This means that it measures the added predictive power of the kth lag in a model containing Y t-1 to Y t-k terms as the regressors. The last two columns reported in the correlogram are the Ljung-Box Q-statistics and their p-values. 2 m ρ Ljung-Box (LB) = n(n + 2) k k=1 n k ~χ2 m The Q-statistics test the joint hypothesis that all the ρk up to certain lags are simultaneously equal to zero. if the computed Q exceeds the critical Q value, in the Chi Square distribution, reject the Oct 22-28, null 2016 hypothesis that all the (true) ρks are zero; at least some of them must be nonzero. 32 Delivered by Dr. Nathaniel E. Urama

The correlogram of the ARIMA Model The correlogram view compares the autocorrelation pattern of the structural residuals and that of the estimated model The structural residuals are the residuals after removing the effect of the fitted exogenous regressors but not the ARIMA terms For a properly specified model, the residual and theoretical (estimated) autocorrelations and partial autocorrelations should be close. Steps in E-view: From the ARIMA Diagnostic Views dialog box, select correlogram, enter number of lags, graph, OK, to display the graph or, table, OK, to displays the table Delivered by Dr. Nathaniel E. Urama 33

Autocorrelation Function (ACF) and Correlogram How does a sample correlogram enable us to find out if a particular series is AR(p) MA(p) or ARMA(p,q)? Delivered by Dr. Nathaniel E. Urama 34

BJ Identification Methods: An e.g. AC and PAC of AR(1), MA(1) and ARMA(1, 1) and ARIMA(1, 1,1) for the Annual OER Septs for AR(1) in Eviews: Quick, estimate equation, enter the equation as oer c ar 1, ok. Or d(oer) c ar 1, Ok if the series is I(1). On the equation dialogue box (DB), click on View, ARMA structure. On the ARMA Diagnostic views DB, click Correlogram, graph, ok. 1 35 Delivered by Dr. Nathaniel E. Urama

AC and PAC of AR(1) process of Annual OER From E-views 9 Delivered by Dr. Nathaniel E. Urama 36

AC and PAC of MA(1) process of OER Delivered by Dr. Nathaniel E. Urama 37

AC and PAC of ARMA(1, 1) process of OER From E-views 9 Delivered by Dr. Nathaniel E. Urama 38

AC and PAC of ARMA(1, 1, 1) process of Annual OER Delivered by Dr. Nathaniel E. Urama 39

AC and PAC of AR(1) process of Quarterly OER From E-views 8 From E-views 9 Delivered by Dr. Nathaniel E. Urama 40

AC and PAC of MA(1) process of Quarterly OER Delivered by Dr. Nathaniel E. Urama 41

AC and PAC of ARMA(1,1,1) process of Quarterly OER From E-views 8 Delivered by Dr. Nathaniel E. Urama 42

ARIMA (p, d, q) Model Selection: Information Criteria Since the actual data ACF and PACF rarely exhibits the theoretical pattern shown, identification will be difficult using them An alternative is to use the information criteria namely: Akaike (AIC), Schwarz (SBIC) and Hannan-Queen (HQIC). AIC = ln σ 2 SBIC = ln σ 2 HQIC = ln σ 2 + 2k T + k ln T T + 2k T ln(ln( T)) Where σ 2 is the residual variance = RSS T The objective is the select the model with the least IC 43 Delivered by Dr. Nathaniel E. Urama

ARIMA (p, d, q) Selection: Information Criteria Cont. Which criterion should be preferred if they suggest different model order? SBIC embodies much stiffer penalty term than the AIC. SBIC is strongly consistent but inefficient. AIC is not consistent but generally efficient. Hence, SBIC asymptotically gives the correct model order than the AIC. However, the average variation in selected model orders from different samples within a given population is greater with SBIC than AIC In conclusion then, no criterion is absolutely superior to others. 44 Delivered by Dr. Nathaniel E. Urama

BJ Identification Methods: An e.g. Cont. Since the ACFs and PACFs didn t show the exact resemblance of of any of the theoretical AR(p), MA(q), nor ARMA(p, d, q), though closer to ARIMA(1, 1, 1) we resort to the Information criteria method. The Information criteria involves estimating different models and comparing their IC values. Tedious!!! testing up to ARMA(p, q) means running (P + 1)X(Q + 1) Models for non seasonal series and(p + 1)X(Q + 1)X4 models for Quarterly series with seasonal pattern. Thanks to E-views for simplifying the case. Steps in Eview 8 and above: Double click the series, Proc. Add-ins, Automatic ARIMA selection, Select the max AR,and MA terms and the IC, OK. For this particular series, the result is as shown below 45 This shows ARIMA( 1, 1, 1) with seasonal pattern in the 4 th lag of the moving average Delivered by Dr. Nathaniel E. Urama

SBIC selection for ARMA(P,Q) for p and q up to 3 each Model Selection Criterion Model Selection Criterion Model Selection Criterion Model Selection Criterion Model Selection Criterion 1 (0,0)(0,0) 5.065968 30 (0,3)(0,4) 3.905965 59 (1,2)(0,0) 4.230029 88 (2,1)(4,0) 4.112137 117 (3,0)(4,4) 4.114159 2 (0,0)(0,0) 5.065968 31 (0,3)(0,0) 3.875592 60 (1,2)(0,4) 3.906879 89 (2,1)(4,0) 4.112137 118 (3,1)(0,0) 4.312175 3 (0,0)(0,4) 5.065968 32 (0,3)(0,0) 3.875592 61 (1,2)(4,0) 4.115243 90 (2,1)(4,4) 3.928206 119 (3,1)(0,0) 4.312175 4 (0,0)(0,0) 5.065968 33 (0,3)(0,4) 3.905965 62 (1,2)(4,0) 4.115243 91 (2,2)(0,0) 4.162886 120 (3,1)(0,4) 3.928491 5 (0,0)(0,0) 5.065968 34 (0,3)(4,0) 3.875592 63 (1,2)(4,4) 3.936816 92 (2,2)(0,0) 4.162886 121 (3,1)(0,0) 4.312175 6 (0,0)(0,4) 5.065968 35 (0,3)(4,0) 3.875592 64 (1,3)(0,0) 3.906967 93 (2,2)(0,4) 3.937926 122 (3,1)(0,0) 4.312175 7 (0,0)(4,0) 5.065968 36 (0,3)(4,4) 3.905965 65 (1,3)(0,0) 3.906967 94 (2,2)(0,0) 4.162886 123 (3,1)(0,4) 3.928491 8 (0,0)(4,0) 5.065968 37 (1,0)(0,0) 4.266771 66 (1,3)(0,4) 3.938244 95 (2,2)(0,0) 4.162886 124 (3,1)(4,0) 4.142421 9 (0,0)(4,4) 5.065968 38 (1,0)(0,0) 4.266771 67 (1,3)(0,0) 3.906967 96 (2,2)(0,4) 3.937926 125 (3,1)(4,0) 4.142421 10 (0,1)(0,0) 4.437998 39 (1,0)(0,4) 4.266771 68 (1,3)(0,0) 3.906967 97 (2,2)(4,0) 4.142343 126 (3,1)(4,4) 3.959461 11 (0,1)(0,0) 4.437998 40 (1,0)(0,0) 4.266771 69 (1,3)(0,4) 3.938244 98 (2,2)(4,0) 4.142343 127 (3,2)(0,0) 4.119912 12 (0,1)(0,4) 4.466906 41 (1,0)(0,0) 4.266771 70 (1,3)(4,0) 3.937242 99 (2,2)(4,4) 3.959457 128 (3,2)(0,0) 4.119912 13 (0,1)(0,0) 4.437998 42 (1,0)(0,4) 4.266771 71 (1,3)(4,0) 3.937242 100 (2,3)(0,0) 3.938342 129 (3,2)(0,4) 3.959675 14 (0,1)(0,0) 4.437998 43 (1,0)(4,0) 4.063103 72 (1,3)(4,4) 4.489726 101 (2,3)(0,0) 3.938342 130 (3,2)(0,0) 4.119912 15 (0,1)(0,4) 4.466906 44 (1,0)(4,0) 4.063103 73 (2,0)(0,0) 4.279902 102 (2,3)(0,4) 3.959699 131 (3,2)(0,0) 4.119912 16 (0,1)(4,0) 4.437998 45 (1,0)(4,4) 4.063103 74 (2,0)(0,0) 4.279902 103 (2,3)(0,0) 3.938342 132 (3,2)(0,4) 3.959675 17 (0,1)(4,0) 4.437998 46 (1,1)(0,0) 4.284612 75 (2,0)(0,4) 4.279902 104 (2,3)(0,0) 3.938342 133 (3,2)(4,0) 4.057974 18 (0,1)(4,4) 4.466906 47 (1,1)(0,0) 4.284612 76 (2,0)(0,0) 4.279902 105 (2,3)(0,4) 3.959699 134 (3,2)(4,0) 4.057974 19 (0,2)(0,0) 4.445625 48 (1,1)(0,4) 3.875515 77 (2,0)(0,0) 4.279902 106 (2,3)(4,0) 3.968636 135 (3,2)(4,4) 3.990809 20 (0,2)(0,0) 4.445625 49 (1,1)(0,0) 4.284612 78 (2,0)(0,4) 4.279902 107 (2,3)(4,0) 3.968636 136 (3,3)(0,0) 3.96972 21 (0,2)(0,4) 4.404162 50 (1,1)(0,0) 4.284612 79 (2,0)(4,0) 4.089123 108 (2,3)(4,4) 3.99069 137 (3,3)(0,0) 3.96972 22 (0,2)(0,0) 4.445625 51 (1,1)(0,4) 3.875515 80 (2,0)(4,0) 4.089123 109 (3,0)(0,0) 4.286892 138 (3,3)(0,4) 3.991068 23 (0,2)(0,0) 4.445625 52 (1,1)(4,0) 4.089922 81 (2,0)(4,4) 4.089123 110 (3,0)(0,0) 4.286892 139 (3,3)(0,0) 3.96972 24 (0,2)(0,4) 4.404162 53 (1,1)(4,0) 4.089922 82 (2,1)(0,0) 4.294018 111 (3,0)(0,4) 4.286892 140 (3,3)(0,0) 3.96972 25 (0,2)(4,0) 4.445625 54 (1,1)(4,4) 3.905451 83 (2,1)(0,0) 4.294018 112 (3,0)(0,0) 4.286892 141 (3,3)(0,4) 3.991068 26 (0,2)(4,0) 4.445625 55 (1,2)(0,0) 4.230029 84 (2,1)(0,4) 3.897261 113 (3,0)(0,0) 4.286892 142 (3,3)(4,0) 4.000031 27 (0,2)(4,4) 4.404162 56 (1,2)(0,0) 4.230029 85 (2,1)(0,0) 4.294018 114 (3,0)(0,4) 4.286892 143 (3,3)(4,0) 4.000031 Delivered by Dr. Nathaniel E. Urama 46 28 (0,3)(0,0) 3.875592 57 (1,2)(0,4) 3.906879 86 (2,1)(0,0) 4.294018 115 (3,0)(4,0) 4.114159 144 (3,3)(4,4) 4.011872 29 (0,3)(0,0) 3.875592 58 (1,2)(0,0) 4.230029 87 (2,1)(0,4) 3.897261 116 (3,0)(4,0) 4.114159

Estimating the ARIMA Model Quick Estimate equation Select LS- Least Square(NLS and ARMA) Type D(OER) ar(1) ma(1) Sma(4) Delivered by Dr. Nathaniel E. Urama 47

Result of ARIMA(1, 1, 1) SMA(4) Model Delivered by Dr. Nathaniel E. Urama 48

AC and PAC for ARIMA(1, 1, 1) SMA(4) Model Delivered by Dr. Nathaniel E. Urama 49

Class Activity 7 7.1 Determine the values of p, d and q ARIMA (p, d, q) model of the following Nigerian series: CPI, INF, RIR., Ycul, using, 1. Correlagram 2. Information Criteria 7.2 Estimate the ARIMA (p, d, q) model of the above Nigerian series and use the AC and PAC to check the correctness f the Model Order. Delivered by Dr. Nathaniel E. Urama 50

ARMA Diagnostic Checking. Over fitting : Deliberately fitting a larger model than that identified to see if all other extra terms would be insignificant. Need for Parsimonious Model Inclusion of irrelevant variables will increase the coefficient standard error and decrease the parameter significance. Whether this happens of not depends on how much the RSS falls and the size of T(number of observations) relative to K(number of Parameters). If T is large, relative to K, the decrease in RSS will outweigh the decrease in Degrees of freedom (T-K) and hence, coefficient standard error will increase. In line with Brooks, Models that are profligate might be inclined to fit the 51 data specific features, which would not be replicated out-of-sample. Delivered by Dr. Nathaniel E. Urama

ARIMA Diagnostic Check: Overfitting The added ar(2) and ar(3) are all insignificant. Delivered by Dr. Nathaniel E. Urama 52

ARMA Diagnostic Checking Cont. Residual diagnostics: Checks the residual for evidence of linear dependence which, if present, suggest that the model is inadequate. The correlogram ( ACF and PACF) If the acf and pacf confirms stationarity of the residual, the model is correctly specified, Steps in E-view: View/Residual Diagnostics/Correlogram-Qstatistic Delivered by Dr. Nathaniel E. Urama 53

ARIMA Diagnostic Check: Correlogram Delivered by Dr. Nathaniel E. Urama 54

ARMA Diagnostic Checking Cont. Residual diagnostics through the serial Correlation LM test If the LM test fail to reject the null hypothesis of no serial Correlation, the model is correctly specified. Steps in E-view: Ensure that ML is bot the Used ARMA method. In Estimate DB, click option and ensure that ML is not the selected method View/Residual Diagnostics/Serial Correlation LM Test 55 Delivered by Dr. Nathaniel E. Urama

ARIMA Diagnostic Check: LM Test Delivered by Dr. Nathaniel E. Urama 56

Notes on BJ diagnostic tests It essentially involves only autocorrelation tests I can only find under and not over parameterized model(brooks. 2008, P.231) 57 Delivered by Dr. Nathaniel E. Urama

Further ARIMA Equation Diagnostics The Stationarity and the Invertibility condition The Stationarity Condition Needed to avoid the previous values of the error term having nondeclining effect on the current values of Y t as the time progresses. Given a general ARMA model in the lag polynomial form α L and θ L as α L Y t = θ L ε t The reported roots are the inverse roots of the characteristics polynomials: α x 1 = 0, and θ x 1 = 0 58 Which may be imaginary, but should have modulus not greater than 1 Delivered by Dr. Nathaniel E. Urama

Further ARIMA Diagnostic Tests Cont. The above is same as the roots of the characteristics equation, 1 α 1 L 1 α 2 L 2 α p L p = 0 all being outside the unit circle. (see Brooks, 2008, p.216) The roots view displays the inverse roots of the AR and/or MA characteristic polynomial, either as a graph or as a table. The graph view plots the inverse roots in the complex plane the horizontal axis is the real part and the vertical axis is the imaginary part of each root. Delivered by Dr. Nathaniel E. Urama 59

Further ARIMA Diagnostic Tests Cont. If the estimated ARMA process is (covariance) stationary, all the AR inverse roots should be inside the unit circle. If the estimated ARMA process is invertible, all MA inverse roots should be inside the unit circle. If the AR has a real Inverse roots with r > 1, a pair of complex reciprocal roots with modulus greater than one, it means that the autoregressive process is explosive. If the MA has reciprocal roots outside the unit circle, we say that the MA process is noninvertible, which makes interpreting and using the MA results difficult. Delivered by Dr. Nathaniel E. Urama 60

Further ARIMA Diagnostic Tests Cont. If the MA has roots with Modulus close to 1, It is a sign that we have over differenced the series. It makes estimation and forecasting difficult. According to Hamilton (1994a, p. 65) however, noninvertibility is not a serious problem since there is always an equivalent representation of the MA model with the inverse roots inside the unit circle. The Iteration should continue with different staring values until invertibility is achieved. Delivered by Dr. Nathaniel E. Urama 61

Further ARIMA Diagnostic Tests Cont. Steps in E-view: On the ARIMA Equation DB, click view, ARMA structure and the ARMA diagnostic views below will appear Select Roots, Graph, ok and the graph in the next slide will appear. Delivered by Dr. Nathaniel E. Urama 62

The Inverse Roots Diagnostic test for ARIMA Delivered by Dr. Nathaniel E. Urama 63 All the Inverse roots of both the AR and MA lies inside the unit circle. The AR is stationary and the MA is ivertible

The Inverse Roots of AR/MA in a Tabular form Both the AR and the MA has no root outside the unit circle. Hence, AR is stationary and MA is invertible. However, the MA has roots close to 1, a sign that we may have over differenced the Delivered by Dr. Nathaniel E. Urama 64 series.

The Inverse Roots Diagnostic test for ARMA Reducing the number of differencing by 1 gives us: an ARIMA(1, 0, 1) model or an ARMA(1, 1) model with the result as below: Delivered by Dr. Nathaniel E. Urama 65

Inverse roots of the AR/MA Polynomial for the ARMA(1,1) All the Inverse roots of both the AR and MA lies inside the unit circle. Delivered by Dr. Nathaniel E. Urama 66

Inverse roots of the AR/MA Polynomial for the ARMA(1,1) All the Inverse roots of both the AR and MA still lies inside the unit circle but the roots of the MA are now far from 1. 67 How ever, since the value of the IC criteria are all smaller for the differenced series, we will adopt it. Delivered by Dr. Nathaniel E. Urama

Impulse Response of an ARIMA model Traces the response of the ARIMA part of the estimated equation to one time shock in the innovation. The accumulated response is the response to step impulse where the same shock occurs in every period from the first. If the estimated ARMA model is stationary, the impulse responses will asymptote to zero, while the accumulated responses will asymptote to its longrun value. Steps in E-view: From the ARIMA Diagnostic Views dialog box, select Impulse Response, enter number of periods, graph, OK, to display the graph or, table, 68 OK, to display the table. Delivered by Dr. Nathaniel E. Urama

Impulse Response of our ARIMA model Delivered by Dr. Nathaniel E. Urama 69

Class Activity 8 Carry out a diagnostic check on the ARIMA models you estimated during Class activity 7 and interpret them. Conduct an impulse response on all of them. End of day 6 Assignment: Read Brooks section 5.9 and adapt the model there to test if Uncovered Interest rate Parity held in Nigeria between 1980 and 2015. Oct 22-28, 2016 70 Delivered by Dr. Nathaniel E. Urama

Forecasting with ARIMA and ARIMAX models In-sample vs Out-of Sample Dynamic vs static Forecast Accuracy Decomposition of the Forecast Errors Class Activity 9: Day 7: Course Details Multivariate models VAR, Parsimonious VAR (PVAR), VEC, ARDL Series stationarity and the VAR model Stability and Stationarity in VAR VAR lag length Selection, Impulse Response and Variance Decomposition Forecasting with VAR In-sample vs Out-of Sample Dynamic vs static VAR diagnostic Checking Class activity 10: Delivered by Dr. Nathaniel E. Urama 71

Forecasting in Econometrics What is Forecasting? The role of forecasting in Economics and Policy Decision Forecasting Accuracy Vs Statistical Significance of a model Structural Forecasting: Forecasting from Parametric/theoretical models return predictions derived from arbitrage pricing models, long-term exchange rate prediction based on purchasing power parity or uncovered interest parity theory. 72 Delivered by Dr. Nathaniel E. Urama

Forecasting in Econometrics Time series Forecasting: Forecasting from a-theoretic/non-parametric model. forecast the future values of a series given its previous values and/or previous values of an error term. Is there a clear distinction? Which one is better? Point and Interval Forecasts. In-sample vs Out-of Sample forecasts Dynamic Vs static forecasts Rolling Vs Recursive forecasts 73 Delivered by Dr. Nathaniel E. Urama

Dynamic Vs Static forecasts The dynamic and static is all about how the lagged value of Y that appears on the right-hand side of the equation should be evaluated Dynamic Calculates, multi-step forecasts from the first period in the forecast sample. From an AR p model Y t = α 0 + α 1 Y t 1 + α 2 Y t 2 + + α p Y t p + ε t The Forecast for Y t is gotten from the actual values of the lagged Y: p Y t = α 0 + i=1 α i Y t i Then forecast of Y t+s uses previously forecasted values for the lagged dependent variables. s Y t+s = α 0 + i=1 p α i Y t+s i + i=s+1 α i Y t+s i 74 Delivered by Dr. Nathaniel E. Urama

Dynamic Vs Static forecasts Dynamic Forecast : Is true multi-step and recursive forecasts. It uses the recursively computed forecast of the lagged value of the dependent variable. requires that data for the exogenous variables be available for every observation in the forecast sample, only available when the estimated equation contains dynamic components that values for any lagged dependent variables be observed at the start of the forecast sample In Dynamic Forecast, any missing values for the explanatory variables will generate an NA for that observation and in all subsequent observation (E-views User guide) 75 Delivered by Dr. Nathaniel E. Urama

Static Forecast The Static Forecast calculates a sequence of one-step ahead forecasts, using the actual, rather than forecasted values for lagged dependent variables, if available. From an AR p model Y t = α 0 + α 1 Y t 1 + α 2 Y t 2 + + α p Y t p + ε t The Forecast for both Y t and Y t+s are gotten from the actual values of the lagged Y if available: p Y t+s = α 0 + i=1 α i Y t+s i 76 Delivered by Dr. Nathaniel E. Urama

Rolling Vs Recursive forecasts Rolling Fix the length of the in-sample period used to estimate the model so that the start and the end date increases successively by one observation. Recursive Fix the initial estimation date but additional observation are added one at a time to the estimation period (Brooks, 2008, p. 246) 77 Delivered by Dr. Nathaniel E. Urama

Forecasting with MA(q) model A moving average process has a memory only of length q, and this limits the sensible forecasting horizon Given an MA q : Y t = θ 0 + ε t + θ 1 ε t 1 + θ 2 ε t 2 + + θ q ε t q * With the assumption of constant Parameter, What hold at time t will also hold at time t + s. Hence, Y t+1 = θ 0 + ε t+1 + θ 1 ε t + θ 2 ε t 1 + + θ q ε t q+1 ** Y t+2 = θ 0 + ε t+2 + θ 1 ε t+1 + θ 2 ε t + + θ q ε t q+2 *** 78 Y t+q = θ 0 + ε t+q + θ 1 ε t+q 1 + θ 2 ε t+q 2 + + θ q ε t q+q Delivered by Dr. Nathaniel E. Urama

Forecasting with MA(q) model Cont. The forecasts of Y t+1 given as f t,1 is gotten thus: f t,1 = E(Y t+1 Ωt ) = E θ 0 + ε t+1 + θ 1 ε t + θ 2 ε t 1 + + θ q ε t q+1 Ω t Where E(Y t+1 Ωt ) means the expected value of Y at time t + 1 given all the information available up to and including time t. Since E(ε t+s Ωt ) = 0 s > 0, f t,1 = E(Y t+1 Ωt ) = θ 0 + θ 1 ε t + θ 2 ε t 1 + + θ q ε t q+1 f t,q = E(Y t+q Ωt ) = θ 0 + θ q ε t f t,s = E(Y t+s Ωt ) = θ 0 s q + 1 What the last two equations show is that the MA(q) process has a memory of only q periods and all forecasts above q periods ahead collapse to the intercept if there is intercept, 0 otherwise. Delivered by Dr. Nathaniel E. Urama 79

Forecasting with AR(p) model Unlike a moving average process, an autoregressive process has infinite memory (see Brooks, 2008, p.249-250). (Explain this) Given an AR p : Y t = α 0 + α 1 Y t 1 +α 2 Y t 2 + + α p Y t p + ε t With the assumption of constant Parameter, What hold at time t will also hold at time t + s. Hence, Y t+1 = α 0 + ε t+1 + α 1 Y t +α 2 Y t 1 + + α q Y t p+1 Y t+2 = α 0 + ε t+2 + α 1 Y t+1 +α 2 Y t + + α q Y t p+2 Y t+p = α 0 + ε t+p + α 1 Y t+p 1 +α 2 Y t+p 2 + + α p Y t p+p 80 Delivered by Dr. Nathaniel E. Urama

Forecasting with AR(p) model Cont. The forecasts of Y t+1 given as f t,1 is : f t,1 = E(Y t+1 Ωt ) = E α 0 + ε t+1 + α 1 Y t + α 2 Y t 1 + + α p Y t p+1 Ω t = α 0 +α 1 Y t + α 2 Y t 1 + + α p Y t p+1 f t,2 = E(Y t+2 Ωt ) = E α 0 + ε t+2 + α 1 Y t+1 + α 2 Y t + + α p Y t p+2 Ω t = α 0 + α 1 E Y t+1 t + α 2 E Y t + + α p E Y t p+2 = α 0 + α 1 f t,1 + α 2 y t + + α p Y t p+2 f t,p = E(Y t+p Ωt ) = E α 0 + ε t+p + α 1 Y t+p 1 + α 2 Y t+p 2 + + α p Y t+p p Ω t = α 0 + α 1 E Y t+p 1 t + α 2 E Y t+p 2 + + α p E Y t+p p = α 0 + α 1 f t,p 1 + α 2 f t,p 2 + + α p Y t Etc. These confirms that the AR(p) process has infinite memory 81 Delivered by Dr. Nathaniel E. Urama

Forecasting with ARIMA models Let f t,s be a forecast from ARMA(P, q) model at time t for s steps into the future. p Then, f t,s = i=1 q α i f t,s i + j=1 θ j ε t+s j The above is known as Forecast function Where f t,s = y t+s,s 0 ; and ε t+s = 0, s > 0 = ε t+s, S 0 α i and θ j are the autoregressive and moving average coefficients, respectively (Brooks, 2008, p. 249) The value of s (the forecast horizon) is determined by the ARIMA structure. Delivered by Dr. Nathaniel E. Urama 82

Forecasting OER with AR(1) process in E-view Estimate D(OER) with Ar(1) process: In the Command window, Click Quick, estimate equation In the equation estimation DB, type: D(oer) c ar(1) Since our workfile data rage is 1972Q3 to 2012Q4, for us to be able to obtain out-ofsample forecast, we reduce the sample range in the equation estimation DB to : 1972Q3 to 2011Q4, click ok. In the result window, click forecast. The Forecast Db will appear. In the Forecast DB, Select the series to forecast (will this be in all cases?), type the name for the forecast series, Select the method, and the forecast sample, ok Delivered by Dr. Nathaniel E. Urama 83

Dynamic Forecasting OER with AR(1) process in E-view 800 600 400 200 0-200 Forecast: OER_IN_DF Actual: OER Forecast sample: 1972Q3 2012Q4 Included observations: 162 Root Mean Squared Error 39.18590 Mean Absolute Error 30.47683 Mean Abs. Percent Error 1451.679 Theil Inequality Coefficient 0.239694 Bias Proportion 0.510870 Variance Proportion 0.119438 Covariance Proportion 0.369691-400 1975 1980 1985 1990 1995 2000 2005 2010 OER_IN_DF ± 2 S.E. 84 Delivered by Dr. Nathaniel E. Urama

Static Forecasting OER with AR(1) process in E-view 170 160 150 140 130 120 Forecast: OER_OUT_SF Actual: OER Forecast sample: 2005Q3 2012Q4 Included observations: 30 Root Mean Squared Error 2.015518 Mean Absolute Error 0.956700 Mean Abs. Percent Error 0.713574 Theil Inequality Coefficient 0.007236 Bias Proportion 0.000000 Variance Proportion 0.026305 Covariance Proportion 0.973694 110 2005 2006 2007 2008 2009 2010 2011 2012 85 OER_OUT_SF Delivered by Dr. Nathaniel E. Urama ± 2 S.E.

Forecasting OER with AR(1) process in E-view To see the difference between dynamic and Static static forecast, increase the workfile Dynamic range to 2013Q4 but having data only up to 2012Q4 and try to forecast to 2013Q4. Delivered by Dr. Nathaniel E. Urama 86

162 Forecasting OER with AR(1) process in E-view 160 158 156 154 152 150 IV I II III IV I II III IV I II III IV 2010 2011 2012 2013 Delivered by Dr. Nathaniel E. Urama 87 OER OER_AR_DF OER_AR_SF

Forecasting OER with MA(1) process in E-view Estimate D(OER) with MA(1) process: In the Command window, Click Quick, estimate equation In the equation estimation DB, type: D(oer) c ma(1) sma(4) reduce the sample range in the equation estimation DB to 1972Q3 to 2011Q4, click ok. Delivered by Dr. Nathaniel E. Urama 88

Forecasting OER with MA(1) process in E-view The dynamic Dynamic forecast covers all the forecast horizon Staticwhile the static forecast only one period above the available data 89 Delivered by Dr. Nathaniel E. Urama

162 Forecasting OER with MA(1) process in E-view 160 158 156 154 152 150 148 IV I II III IV I II III IV I II III IV 2010 2011 2012 2013 OER OER_MA_DF OER_MA_SF 90 Delivered by Dr. Nathaniel E. Urama

162 Forecasting OER with MA(1) process in E-view 160 158 156 154 152 150 148 IV I II III IV I II III IV I II III IV 2010 2011 2012 2013 OER OER_AR_SF OER_MA_SF OER_AR_DF OER_MA_DF Delivered by Dr. Nathaniel E. Urama 91

Forecasting OER with ARIMA(1,1,1) process in E-view Table of ARIAM forecast result 92 Delivered by Dr. Nathaniel E. Urama

62 Forecasting OER with ARIMA(1,1,1) process in E-view 60 58 56 54 Gaph of ARIAM forecast result 52 50 48 IV I II III IV I II III IV I II III IV 2010 2011 2012 2013 OER OER_ARIMA_DF OER_ARIMA_SF Delivered by Dr. Nathaniel E. Urama 93

Forecast Accuracy The level of accuracy of a forecast is determined using the forecast error (FE). The FE for a given observation is the difference between the actual value and its forecast. It will therefore be positive if the forecast is too low and negative if the forecast is too high. Thus, before the forecast errors are aggregated, they are usually squared or the absolute value taken (Brooks, 2008 p. 252-253) as in the next slide. Delivered by Dr. Nathaniel E. Urama 94

Forecast Accuracy criteria Mean Square Error MSE = 1 T T1 1 T t=t1 Y t+s f t,s 2 Mean Absolute Error MAE = 1 T T1 1 T t=t1 Y t+s f t,s Mean Absolute % Error MAPE = 100 T T1 1 T t=t1 Y t+s f t,s Y t+s Adjusted MAPE AMAPE = 100 T T1 1 T t=t1 Y t+s f t,s Y t+s +f t,s T is total sample size (in-sample + out-of-sample) T 1 is the first out-of-sample forecast observation. 95 Delivered by Dr. Nathaniel E. Urama

Decision with the forecast accuracy result MSE, MAE and MAPE can not be used to judge a single forecast: They are unbounded from above. The only way is to compare results from different models for same data and forecast period and choose the model with the lowest value Which of the criterion is better? Use MSE when large forecast errors are disproportionately more serious than smaller errors Authors like Makridakis (1993, p. 528) see MAPE as the best. Brooks (2008, p. 253) argues that compared to MAPE, AMAPE is better(except when the series and the forecasts can take on opposite signs) as, it corrects for the problem of asymmetry between the actual and forecast values If they Y t+s and the f t,s are equal and opposite, the denominator in AMAPE turns to 0 Delivered by Dr. Nathaniel E. Urama 96

Theil s U-statistic Theil s U-statistic is another criterion for judging forecast accuracy. It is given as: U = t=t T 1 T t=t1 Y t+s f 2 t,s Y t+s Y t+s fb 2 t,s Y t+s fb t,s is the forecast obtained from a benchmark model (typically a simple model such as a naive or random walk). U = 1 implies that both the model and the benchmark are equally in(accurate) U < 1 implies that the model is superior to the benchmark U > 1 implies that the model is inferior to the benchmark Delivered by Dr. Nathaniel E. Urama 97

Decomposition of the forecast errors The mean squared forecast error can be decomposed into: a bias proportion, a variance proportion and a covariance proportion. The bias component measures the extent to which the mean of the forecasts is different from the mean of the actual data the variance component measures the difference between the variation of the forecasts and the variation of the actual data, while the covariance component captures any remaining unsystematic part of the forecast errors. Accurate forecasts would be unbiased and also have a small variance proportion, so that most of the forecast error should be attributable to the covariance, see Pindyck and Rubinfeld (1998, p. 210-214). Delivered by Dr. Nathaniel E. Urama 98

Statistical versus financial or economic loss functions. The statistical loss functions vs practical usefulness. Models with lower forecast error may not necessarily be more profitable when used in business forecasting Rather, Leitch and Tanner, (1991) as cited in Brooks (2008) argue that models that can accurately forecast the sign, or predict turning points in a series are more profitable. These are computed as: % correct sign prediction = For z t+s = 1 if Y t,s f t,s > 0, 0 otherwise 1 T T T 1 1 t=t 1 z t+s % correct direction of change predictions = For z t+s = 1 if Y t,s Y t, f t,s Y t, > 0, 0 otherwise 1 T T T 1 1 t=t 1 z t+s 99 Delivered by Dr. Nathaniel E. Urama

Class activity 9 1. Carry out dynamic and static forecast of 8 period ahead values of the series modeled in class activity 7. 2. Model the same variable with ARMA(1,1) process and compare their statistical loss function 3. Compute both the % correct sign prediction and the % correct direction of change prediction for the two models and compare them. 100 Delivered by Dr. Nathaniel E. Urama