1. Time Series EC 532 2017 Burak Saltoglu

2. Ec532 2nd half: Time Series Analysis
Topic 1 linear time series
Topic 2 nonstationary time series
Topic 3 Cointegration and unit roots
Topic 4 Vector Autoregression (VAR)
Topic 5 Volatility modeling (if time allows)

3. References

4. Main reference book for the course
Vance Martin, Stan Hurn and David Harris, 2013, Econometric Modellig with Time Series
R, Matlab and GAUSS codes are very useful.

5. Time series books Hamiton, J (1994); Time Series Analysis
Enders W (2014), Applied Time Series
Chatfield (2003), The Analysis of Time Series
Diebold F (2006), Elements of Forecasting
9/22/2018

6. References books : Ruey Tsay, 2013
Walters Applied Time Series Methods, Wiley, 2013
Granger Long Run Economic Relationships, 1990.
Hamilton Time Series Analysis, 1994.

7. Topic 1: Linear Time series Outline
Non-stationary time series
Distributed Lag Models
Nonlinear Models

8. outline Linear Time Series Models ARDL Models Granger Causality Test
AR and MA processes
Diagnostics in Time Series
Correlogram
Box-Pierce Q Statistics
Ljung-Box (LB) Statistics
Forecasting

9. Later in topic 2
Stationary versus Non-stationary Times Series
Testing for Stationarity

10. The Reasons for using Time Series
Time Series: consists of a set of observations on a variable, y, taken at equally spaced intervals over time.
Why study time series: 2 reasons analysis and modelling
The aim of the analysis is to summarize the characteristic of time series
Modelling: forecasting the future values.:
Why we rely on time series?
Psychological Reasons:
People do not change their habits immediately
Technological Reasons:
Quantity of a resource needed or bought might not be so adaptive in many cases
Instutitional Reasons:
There might be some limitations on individuals

11. Distributed Lag Models
In the distributed lag (DL) model we have not only current value of the explanatory variable but also its past value(s).
With DL models, the effect of a shock in the explanatory variable lasts more.
We can estimate DL models (in principal)with OLS. Because the lags of X are also non-stochastic.

12. Autoregressive Models
In the Autoregressive (AR) models, the past value(s) of the dependent variables becomes an explanatory variable.
We can not esitmate an autoregressive model with OLS due to
1.Presence of stochastic explanatory variables and
2.Posibility of serial correlation

13. ARDL Models
In the ARDL models, we have both AR and DL part in one regression.

14. Granger Causality Test
Let us consider the relation between GNP and money supply.
A regression analysis can show us the relation between these two.
But our regression analysis can not say us the direction of the relation.
The granger causality test examines the causality between series, the direction of the relation.
We can test whether GNP causes money supply to increase or a monetary expansion lead GNP to rise, under conditions defined by Granger.

15. Granger Causality Test
Steps for testing M (granger) causes GNP;
Regress GNP on all lagged GNP obtain
Regress GNP on all lagged GNP and all lagged M obtain
The null is ’s are alll zero.
Test statistics;
where m number of lags, k the number of parameters in step-2.
df(m,n-k)

16. Granger Causality Test

17. Linear Time Series Models: y(t)
Time series analysis is useful when the economic relationship is difficult to set
Even if there are explanatory variables to express y, it is not possible to forecast y(t)

18. Stationary Stochastic Process
Any time series data can be thought of as being generated by a stochastic process.
A stochastic process is said to be stationary if its mean and variance are constant over time
the value of covariance between two time depends only on the distance or lag between the two time periods and not on the actual time at which the covariance is computed.

19. Times series and white noise
a process is said to be white noise if it follows the following properties

20. Stationary Time Series
If a time series is time invariant with respect to changes in time
The process can be estimated with fixed coefficients
Strict-sense Stationarity:

21. Stationarity
Wide sense stationarity

22. Stationarity
Strict sense stationarity implies wide sense stationarity but the reverse is not true.
İmplication of stationarity: inference we obtain from a non-stationary series is misleading and wrong.

23. Linear Time Series Models-AR
Basic ARMA Models

24. Lag Operators
Or we can use lag polynomials

25. Lag operators and polynomials

26. AR vs MA Representation Wold Decomposition

27. AUTOCORRELATIONS and AUTOCOVARIANCE FUNCTIONS
𝑦 𝑡 = 𝑗=0 ∞ 𝜙 2𝑗 𝜀 𝑡−𝑗
𝑣𝑎𝑟(𝑦 𝑡 )=E 𝑗=0 ∞ 𝜙 2𝑗 𝜀 𝑡−𝑗 2
𝑣𝑎𝑟(𝑦 𝑡 )= 𝑗=0 ∞ 𝜙 2𝑗 E( 𝜀 2 𝑡−𝑗 )
𝑗=0 ∞ 𝜙 2𝑗 = 1+ 𝜙 2 +…= 1 1− 𝜙 2
𝑣𝑎𝑟(𝑦 𝑡 )= 1 1− 𝜙 2 E( 𝜀 2 𝑡−𝑗 ) = 1 1− 𝜙 2 𝜎 2

28. Autocorrelation

29. Sample counterpart of autocovariance function
𝛾 0 = 𝑡=1 𝑇 𝑦 𝑡 − 𝑦 2 = 𝜎 2
𝛾 𝑘 = 𝑡=1 𝑇 𝑦 𝑡 − 𝑦 ( 𝑦 𝑡−𝑘 − 𝑦 )
k=1,…
Because of stationarity:
𝛾 𝑘 = 𝛾 −𝑘
𝜌 𝑘 = 𝛾 𝑘 𝛾 0

30. Partial Autocorrelation
An AR process has
A geometrically decaying acf
Number of non-zero points of pacf=AR order.
MA: geometrically decaying pacf
Number of non-zero points of acf=MA order.
Measures the correlation between an observation k periods ago and the current observation, after controling for intermediate lags.
For The first lags pacf and acf are equal.

31. Linear Time Series -AR
For AR(1);

32. Linear Time Series Models-AR

33. Linear Time Series Models-AR

34. Linear Time Series Models-AR(1)
So if you have a data which is generated by an AR(1) process, it is correlogram will diminish slowly (if it is stationary)

35. 𝑦 𝑡

36. 𝑦 𝑡 =0.95 𝑦 𝑡−1 + ε 𝑡

37. AR(1) process
𝑦 𝑡 =0.99 𝑦 𝑡−1 + ε 𝑡

38. AR process simulation 𝑦 𝑡 =0.90 𝑦 𝑡−1 + ε 𝑡

39. AR process simulation
𝑦 𝑡 =0.5 𝑦 𝑡−1 + ε 𝑡

40. AR(1) with weak predictable part
𝑦 𝑡 =0.05 𝑦 𝑡−1 + ε 𝑡

41. Turkish GDP growth

42. Turkish GDP and ınflation
M =
Turkish GDP and ınflation
GDP (quarterly:
Mean
4.6635
st.deviation
5.6062
Skewness
kurtosis
4.3208

43. US GDP: (Quarterly)

44. Turkish GDP quarterly: Autocorrelations

45. AR(1) Application on Turkish growth rate
Turkish GDP
estimate SE
t-stat
constant
0.93
0.46
2.05
AR(1)
0.80
0.07
11.99
variance
11.86
1.87
6.33
mean
4.6635
st.deviation
5.6062
Skewness
kurtosis
4.3208
𝑣𝑎𝑟(𝑦 𝑡 )= 1 1− 𝜙 2 𝜎 2
𝐸(𝑦 𝑡 )= δ 1− 𝜙 2 = − =4.562

46. Turkish inflation autocorrelations: Persistency

47. AR(1) Application on Turkish growth rate
Turkish inflation
estimate SE
t-stat
constant
1.45
1.74
0.8
AR(1)
0.959
0.02
45.9
variance
78.2
2.8 27
Inflation monthly:
Mean 35
st.deviation 31
Skewness
0.94
kurtosis
3.13
𝐸(𝑦 𝑡 )= δ 1− 𝜙 2 = − =35.36

48. Turkish inflation

49. 𝑦 𝑡 = ε 𝑡

50. 𝑦 𝑡 =0.9 𝑦 𝑡−1 + ε 𝑡
𝑦 𝑡 =-0.8 𝑦 𝑡−1 + ε 𝑡

51. Linear Time Series Models-AR(p)
Autoregressive
Expected value of Y;

52. Linear Time Series Models-MA
Moving Average MA(k) Process
The term ‘moving average’ comes from the fact that y is constructed from a weighted sum of the two most recent Error terms

54. MA(1) Correlogram

55. Linear Time Series Models-MA(1) typo!
So if you have a data that is generated by MA(1) its correlogram will decline to zero quickly (after one lag.)

56. An MA(1) example
One major implication is the MA(1) process has a memory of only one
Lag. i.e.
MA(1) process forgets immediately after one term or only remembers the
Just one Previous realization.

57. Variance-autocovariance MA(2)
**Since it is white noise

58. MA(2)

59. Linear Time Series Models-MA
Moving Average MA(k) Process
Error term is white noise.
MA(k) has k+2 parameters
Variance of y;
Need to derive gamma2,…

60. Homework, derive the autocorrelation function for MA(3),..MA(k).

61. ARMA Models: ARMA(1,1)

62. ARMA(1,1)

63. ARMA(1,1)

65. Maximum likelihood estimation

66. Deriving the likelihood function

67. Log-likelihood

69. Estimation AR(1)
Since T and other parameters are constant we can ignore them in optimization and use

70. Likelihood function for ARMA(1,1) process

71. Model Selection How well does it fit the data?
Adding additional lags for p and q will reduce the SSR.
Adding new variables decrease the degrees of freedom
In addition, adding new variables decreases the forecasting performance of the fitted model.
Parsimonious model: optimizes this trade-off

72. Two Model Selection Criteria
Akaike Information Criterion: Schwartz Bayesian Criterion.
AIC: k is the number of parameters estimated if intercept term is allowed: (p+q+1) else k=p+q.
T: number of observations
Choose the lag order which minimizes the AIC or SBC
AIC may be biased towards selecting overparametrized model whereas SBC is asympoticaly consistent

73. Chararterization of Time Series
Visual inspection
Autocorrelation order selection
Test for significance
Barlett (individual)
Box Ljung (joint)

74. Correlogram Under stationarity,
One simple test of stationarity is based on autocorrelation function (ACF). ACF at lag k is;
Under stationarity,

75. Sample Autocorrelation

76. Correlogram If we plot against k, the graph is called as correlogram.
As an example let us look at the correlogram of Turkey’s GDP.

77. Autocorrelation Function
Correlogram
Autocorrelation Function

78. Test for autocorrelation
Barlett Test: to test for

80. ISE30 Return Correlation

81. Box-Pierce Q Statistics
To test the joint hypothesis that all the autocorrelation coefficients are simultaneously zero, one can use the Q statistics.
where;
m= lag length
T= sample size

82. Ljung-Box (LB) Statistics
It is variant of Q statistics as;

83. Box-Pierce Q Statistics

84. Box Jenkins approach to time series
data
Stop: If the series are non-stationary
Identification
Choose the order of p q ARMA
Estimate ARMA coefficients
Diagnostic checking:
Is the model appropriate
Forecasting

85. forecasting T T+R Today Ex ante period ESTIMATION PERIOD t=1,…T
Ex post forecasting period
T+1,…T+R

86. Introduction to forecasting

88. In practice
If we can consistently estimate the order via AIC then one can forecast the future values of y.
There are alternative measures to conduct forecast accuracy

89. Mean Square Prediction Error Method (MSPE)
Choose model with the lowest MSPE
If there are observations in the holdback periods, the MSPE for Model 1 is defined as:

90. A Forecasting example for AR(1)
Suppose we are given

91. A Forecasting example for AR(1)
Left for forecasting

93. Introduction to forecasting

94. Forecast of AR(1) model forecast actual y(151) forecast -6.452201702
y(152) forecast
y(153) forecast
y(154) forecast
y(155) forecast
y(156) forecast
y(157) forecast
y(158) forecast
y(159) forecast
y(160) forecast
y(161) forecast

95. AR(1) forecast

97. Forecast Combinations
Assume that there are 2 competing forecast model: a and b
In addition, the forecast errors also has the same linear combination
Assuming no correlation between model a and b

98. So, if model a has better prediction error than b we give more weights to a.

99. Forecasting combination
Voting behavior:
suppose company A forecasts the vote for party X: 40%, B forecasts 50%.
past survey performances:
𝜎 2 𝑎=30%
𝜎 2 𝑏=20%

100. Using regression for forecast combinations
Run the following regression and then do the forecasts on the basis of estimated coefficients

101. Summary Find the AR, MA order via autocovariances, correlogram plots
Use, AIC, SBC to choose orders
Check LB stats
Run a regression
Do forecasting (use RMSE or MSE) to choose the best out-of-sample forecasting model.

102. Topic II: Testing for Stationarity and Unit Roots
EC 532

103. Outline What is unit roots? Why is it important? Test for unit roots
Spurious regression
Test for unit roots
Dickey Fuller
Augmented Dickey Fuller tests

104. Stationarity and random walk
Can we test via ACF or Box Ljung?
Why a formal test is necessary?
Source: W Enders Chapter 4, chapter 6

105. Spurious Regression
Regressions involving time series data include the possibility of obtaining spurious or dubious results signals the spurious regression.
Two variables carrying the same trend makes two series to move together
this does not mean that there is a genuine or natural relationship.

106. Spurios regression
One of OLS assumptions was the stationarity of these series we will call such regression as spurious regression (Newbold and Granger (1974)).

107. Unit roots and cointegration
Clive Granger
Robert Engle

108. Spurious regression
the least squares estimates are not consistent and regular tests and inference do not hold.
As rule of thumb (Granger and Newbold,1974)

109. Example Spurious Regression : two simulated RW:Ar1.xls
Xt = Xt-1 + ut ut~N(0,1)
Yt = Yt-1 + εt εt~N(0,1)
ut and εt are independent
Spurious regression:
Yt = βXt + ut
Coefficients
Standard Error
t Stat
P-value
X Variable 1
9.87E-16

110. Examples:Gozalo

111. Unit Roots: Stationarity

112. Stationary and unit roots

113. Some Time Series Models: Random Walk Model
Where error term follows the white noise property with the following properties

114. Random Walk
Now let us look at the dynamics of such a model;
𝜎 2

115. Implications of Random walk

117. Random Walk: BİST30 index

118. Random Walk:ISE percentage returns

120. Why a formal test is necessary?
For instance, daily brent oil series given below graph shows a series non-stationarity time series.

121. Brent Oil:20 years of daily data
End of lecture

122. How instructive to use ACF?

123. Does Crude Oil data follow random walk? (or does it contain unit root)
Neither Graph nor autocovariance functions can be formal proof of the existence of random walk series.
How about standard t-test?

124. Testing for Unit Roots: Dickey Fuller
    
But it would not be appropriate to use this information to reject the null of unit root.
This t-test is not appropriate under the null of a unit –root.
Dickey and Fuller (1979,1981) developed a formal test for unit roots.
Hypothesis tests based on non-stationary variables cannot be analytically evaluated.
But non-standard test statistics can be obtained via Monte Carlo

125. Dickey Fuller Test
These are three versions of the Dickey-Fuller (DF) unit root tests.
The null hypothesis for all versions is same whether beta1 is zero or not.

126. Dickey Fuller Test
These are three versions of the Dickey-Fuller (DF) unit root tests.
The null hypothesis for all versions is same whether beta1 is zero or not.

127. Dickey Fuller Test
The test involves to estimate any of the below specifications

128. Dickey Fuller test
So we will run and test the slope to be significant or not
So the test statistic is the same as conventioanl t-test.

129. Running DF Regression

130. Testing DF in EVIEWS

131. DF: EVIEWS

132. Testing for DF for other specifications: RW with trend

133. Dickey Fuller F-test (1981).
Now of course the test statistic is distributed under F test which can be found in Dickey Fuller tables.
They are calculated under conventional F tests.

134. Dickey Fuller Test
These are three versions of the Dickey-Fuller (DF) unit root tests.
The null hypothesis for all versions is same whether beta1 is zero or not.

135. Augemented Dickey Fuller

136. Augmented Dickey Fuller Test
With Dickey-Fuller (ADF) test we can handle with the autocorrelation problem. The m, number of lags included, should be big enough so that the error term is not serially correlated.
The null hypothesis is again the same. Let us consider GDP example again

137. Augmented Dickey Fuller Test

138. Augmented Dickey Fuller Test
At 99% confidence level, we can not reject the null.
“ not augmented”

139. Augmented Dickey Fuller Test
At 99% confidence level, we reject the null.
This time we “augmented” the regression to handle with serial correlation
***Because GDP is not stationary at level and stationary at first difference,it is called integrated order one, I(1). Then a stationary serie is I(0).

140. Augmented Dickey Fuller Test
In order to handle the autocorrelation problem Augmented Dickey-Fuller (ADF) test is proposed. The p, number of lags included, should be big enough so that the error term is not serially correlated.
So in practice we use either SBC or AIC to clean the residuals.
The null hypothesis is again the same.

141. ADF

142. Example:Daily Brent Oil We can not reject the null of unit root
t-Statistic   Prob.*
Augmented Dickey-Fuller test stat p= 0.8823
Test critical values: 1% level
5% level
10% level
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
Dependent Variable: D(BRENT)
Included observations: 5137 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.  
BRENT(-1) C
@TREND(1) 1.78E E
R-squared     Mean dependent var

143. Diagnostics: Monthly trl30

146. Trl30 and 360

147. I(1) ve I(0) Series
If a series is stationary it is said to be I(0) series
If a series is not stationary but its first difference is stationary it is called to be difference stationary or I(1).

148. Next presentation will investigate the stationarity behaviour of more than one time series known as co-integration.

149. COINTEGRATION
EC332
Burak Saltoglu

150. Economic theory, implies equilibrium relationships between the levels of time series variables that are best described as being I(1).
Similarly, arbitrage arguments imply that the I(1) prices of certain financial time series are linked. (two stocks, two emerging market bonds etc).

151. Cointegration
If two (or more) series are themselves non-stationary (I(1)), but a linear combination of them is stationary (I(0)) then these series are said to be co-integrated.
Examples:
Inflation and interest rates,
Exchange Rates and inflation rates,
Money Demand: inflation, interest rates, income

153. Brent vs wti

154. Crude oil futures

155. Usd treasury 2 year vs 30 years

156. Money demand r:interest rates, y;income, infl: inflation.
Each series in the above eqn may be nonstationary (I(1)) but the money demand relationship may be stationary...
All of the above series may wander around individually but as an equilibrium relationship MD is stable....
Or even though the series themselves may be non-stationary, they will move closely together over time and their difference will be stationary.

157. COINTEGRATION ANALYSIS
Consider the m time series variables y1t, ,y2t,…,ymt known to non-stationary,
ie. suppose
Then, yt=(y1t, y2t,…,ymt)’ are said to form one or more cointegrating relations
if there are linear combinations of yit’s that are I (0) ie.
i.e if there exists an matrix such that
Where, r denotes the number of cointegrating vectors. 16

158. Testing for Cointegration Engle – Granger Residual-Based Tests Econometrica, 1987
Step 1:
Run an OLS regression of y1t (say) on the rest of the variables: namely y2t, y3t, …ymt, and save the residual from this regression 17

159. Dickey Fuller Test
Dickey-Fuller (DF) unit root tests.

160. Residual Based Cointegration test: Dickey Fuller test
Therefore, testing for co-integration yields to test whether the residuals from a combination of I(1) series are I(0).
If u: is an I(0) then we conclude
Even the individual data series are I(1) their linear combination might be I(0).
This means that there is an equilibrium vector and if the variables divert from equilibrium they will converge there at a later date.
If the residuals appear to be I(1) then there does not exist any co-integration relationship implying that the inference obtained from these variables are not reliable.

161. Higher order integration
If two series are I(2) may be they might have an I(1) relationship.

162. Examples of cointegration: brent wti regression
Null Hypothesis: RESID01 has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic - based on SIC, maxlag=12)
t-Statistic   Prob.*
Augmented Dickey-Fuller test statistic  0.0009
Test critical values: 1% level
5% level
10% level

163. Example of ECM
The following is the ECM that can be formed,

164. COINTEGRATION and Error Correction Mechanism
Estimation of the ECM 22

165. Error Correction Term
The error correction term tells us the speed with which our model returns to equilibrium for a given exogenous shock.
It should have a negative signed, indicating a move back towards equilibrium,
a positive sign indicates movement away from equilibrium
The coefficient should lie between 0 and 1, 0 suggesting no adjustment one time period later, 1 indicates full adjustment

166. An Example
Are Turkish interest rates with different maturities (1 month versus 12 months) co-integrated
Step 1: Test for I(1) for each series.
Step 2: test whether two of these series move together in the long-run.
if yes then set up an Error Correction Mechanism.

170. So both of these series are non-stationary i.e I(1)
Now we test whether there exists a linear combination of these two series which is stationary.

171. COINTEGRATION and Error Correction Mechanism22

172. Test for co-integration

173. COINTEGRATION and Error Correction Mechanism
Estimate the ECM 22

175. ECM regression

176. Use of Cointegration in Economic and Finance
Purchasing Power Parity: FX rate differences between two countries is equal to inflation differences. Big Mac etc…
Uncovered Interest Rate Parity: Exchange rate can be determined with the interest rate differentials
Interest Rate Expectations: Long and short rate of interests should be moving together.
Consumption Income
HEDGE FUNDS! (ECM can be used to make money!) 22

177. conlcusion
Test for co-integration via ADF is easy but might have problems when the relationship is more than 2-dimensional (Johansen is more suitable)
Nonlinear co-integration, Near unit roots, structural breaks are also important.
But stationarity and long run relationship of macro time series should be investigated in detail.

178. Vector Autoregression (VAR)
In 1980’s proposed by Christopher Sims
is an econometric model used to capture the evolution and the interdependencies among
multiple economic time series
generalize the univariate AR models
All the variables in the VAR system are treated symmetrically
(by own lags and the lags of all the other variables in the model
VAR models as a theory-free method to estimate economic relationships,
They consitutean alternative to the "identification restrictions" in structural models

179. VECTOR AUTOREGRESSİON

180. Why VAR?
Christoffer Sims, from princeton (nobel prize winner 2011) First VAR paper in 1980

181. VAR Models
In Vector Autoregression specification, all variables are regressed on their
and others lagged values.For example a simple VAR model is or
which is called VAR(1) model with dimension 2

182. VAR Models Generally VAR(p) model with k dimension is
where each Ai is a k*k matrix of coefficients, m and εt is the k*1 vectors.
Furthermore,
No serial correlation but there can
be contemporaneous correlations

183. An Example VAR Models: 1 month 12 months TRY Interest rates monthly
Generally VAR(p) model with k dimension is
where each Ai is a k*k matrix of coefficients, m and εt is the k*1 vectors.
Furthermore,
No serial correlation but there can
be contemporaneous correlations

184. TRL30R TRL360R
TRL30R(-1)    
 ( )  ( )
[ ] [ ]
TRL30R(-2)  
 ( )  ( )
[ ] [ ]
TRL360R(-1)  
 ( )  ( )
[ ] [ ]
TRL360R(-2)  
 ( )  ( )
[ ] [ ]
C    
 ( )  ( )
[ ] [ ]

185. trl30 and trl360 Akaike information criterion -4.089038
 Schwarz criterion

186. Hypothesis testing
To test whether a VAR with a lag order 8 is preferred to a lag order 10

187. VAR Models
Impulse Response Functions: Suppose we want to see the reaction of our
simple initial VAR(1) model to a shock, say ε1=[1,0]’ and rest is 0, where
....