1 Ka-fu Wong University of Hong Kong Modeling Cycles: MA, AR and ARMA Models

2 Unobserved components model of time series According to the unobserved components model of a time series, the series y t has three components y t = T t + S t + C t Time trend Seasonal component Cyclical component

3 The Starting Point Let y t denote the cyclical component of the time series. We will assume, unless noted otherwise, that y t is a zero-mean covariance stationary process. Recall that part of this assumption is that the time series originated infinitely far back into the past and will continue infinitely far into the future, with the same mean, variance, and autocovariance structure. The starting point for introducing the various kinds of econometric models that are available to describe stationary processes is the Wold Representation Theorem (or, simply, Wold’s theorem).

4 Wold’s theorem According to Wold’s theorem, if y t is a zero mean covariance stationary process than it can be written in the form: where the ε’s are (i) WN(0,σ 2 ), (ii) b 0 = 1, and (iii) In other words, each y t can be expressed in terms of a single linear function of current and (possibly an infinite number of) past drawings of the white noise process, ε t. If y t depends on an infinite number of past ε’s, the weights on these ε’s, i.e., the b i ’s must be going to zero as i gets large (and they must be going to zero at a fast enough rate for the sum of squared b i ’s to converge).

5 Innovations ε t is called the innovation in y t because ε t is that part of y t not predictable from the past history of y t, i.e., E(ε t │ y t-1,y t-2,…)=0 Hence, the forecast (conditional expectation) E(y t │ y t-1,y t-2,…) = E(y t │ ε t-1,ε t-2,…) = E(ε t + b 1 ε t-1 + b 2 ε t-2 +… │ ε t-1,ε t-2,…) = E(ε t │ ε t-1,ε t-2,…) + E(b 1 ε t-1 + b 2 ε t-2 +… │ ε t-1,ε t-2,…) = 0 + (b 1 ε t-1 + b 2 ε t-2 +…) = b 1 ε t-1 + b 2 ε t-2 +… And, the one-step ahead forecast error y t - E(y t │ y t-1,y t-2,…) = (ε t + b 1 ε t-1 + b 2 ε t-2 +…)-(b 1 ε t-1 + b 2 ε t-2 +…) = ε t

6 Mapping Wold to a variety of models The one-step-ahead forecast error is y t - E(y t │ y t-1,y t-2,…) = (ε t + b 1 ε t-1 + b 2 ε t-2 +…)-(b 1 ε t-1 + b 2 ε t-2 +…) = ε t Thus, according to the Wold theorem, each y t can be expressed as the same weighted average of current and past innovations (or, 1- step ahead forecast errors). It turns out that the Wold representation can usually be well- approximated by a variety of models that can expressed in terms of a very small number of parameters. the moving-average (MA) models, the autoregressive (AR) models, and the autoregressive moving-average (ARMA) models.

7 Mapping Wold to a variety of models For example, suppose that the Wold representation has the form: for some b, 0 < b < 1. (i.e., b i = b i ) Then it can be shown that y t = by t-1 + ε t which is an AR(1) model.

8 Mapping Wold to a variety of models The procedure we will follow is to describe each of these three types of models and, especially, the shapes of the autocorrelation and partial autocorrelations that they imply. Then, the game will be to use the sample autocorrelation/partial autocorrelation functions of the data to “guess” which kind of model generated the data. We estimate that model and see if it provide a good fit to the data. If yes, we proceed to the forecasting step using this estimated model of the cyclical component. If not, we guess again …

9 Digression – The Lag Operator The lag operator, L, is a simple but powerful device that is routinely used in applied and theoretical time series analysis, including forecasting. The lag operator is defined as follows – Ly t = y t-1 That is, the operation L applied to y t returns y t-1, which is y t “lagged” one period. Similarly, L 2 y t = y t-2 i.e., the operation L applied twice to y t returns y t-2, y t lagged two periods. More generally, L s y t = y t-s, for any integer s.

10 Digression – The Lag Operator Consider the application of the following polynomial in the lag operator to y t : (b 0 +b 1 L+b 2 L 2 +…+b s L s )y t = b 0 y t + b 1 y t-1 + b 2 y t-2 + …+ b s y t-s where b 0, b 1,…,b s are real numbers. We sometimes shorthand this as B(L)y t, where B(L) = b 0 +b 1 L+b 2 L 2 +…+b s L s. Thus, we can write the Wold representation of y t as B(L)ε t where B(L) is the infinite order polynomial in L: B(L) = 1 + b 1 L + b 2 L 2 + … Similarly, suppose y t = by t-1 + ε t, we can write B(L)y t =ε t, B(L)=1-bL.

11 Moving Average (MA) Models If y t is a (zero-mean) covariance stationary process, then Wold’s theorem tells us that y t can expressed as a linear combination of current and past values of a white noise process, ε t. That is: where the ε’s are (i) WN(0,σ 2 ), (ii) b 0 = 1, and (iii) Suppose that for some positive integer q, it turns out that b q+1, b q+2,… are all equal to zero. That is suppose that y t depends on current and only a finite number of past values of ε: This is called a q-th order moving average process (MA(q))

12 Realization of two MA(1) processes y t = ε t + θε t-1

13 MA(1): y t = ε t + θε t-1 [= (1+θL)ε t ] 1.E(y t )=E(ε t + θε t-1 )= E(ε t )+ θE(ε t-1 )=0 2.Var(y t ) = E[(y t -E(y t )) 2 ]=E(y t 2 ) = E[(ε t + θε t-1 ) 2 ] = E(ε t 2 ) + θ 2 E(ε t-1 2 ) + 2θE(ε t ε t-1 ) = σ 2 + θ 2 σ (since E(ε t ε t-1 ) = E[E(ε t |ε t-1 ) ε t-1 ]=0 ) = (1+ θ 2 )σ 2

14 MA(1): y t = ε t + θε t-1 [= (1+θL)ε t ]  (1) =Cov(y t,y t-1 ) = E[(y t -E(y t ))(y t-1 -E(y t-1 ))] = E(y t y t-1 ) = E[ (ε t + θε t-1 )(ε t-1 + θε t-2 )] = E[ε t ε t-1 + θε t θε t ε t-2 + θ 2 ε t-1 ε t-2 ] = E(ε t ε t-1 )+E(θε t-1 2 )+E(θε t ε t-2 )+E(θ 2 ε t-1 ε t-2 ) = 0 + θσ = θσ 2 4.ρ(1) = Corr(y t,y t-1 ) =  (1)/  (0) = θσ 2 /[(1+ θ 2 )σ 2 ] = θ / (1+ θ 2 ) ρ(1) > 0 if θ > 0 and is < 0 if θ < 0.

15 MA(1): y t = ε t + θε t-1 [= (1+θL)ε t ]  (2) =Cov(y t,y t-2 ) = E[(y t -E(y t ))(y t-2 -E(y t-2 ))] = E(y t y t-2 ) = E[ (ε t + θε t-1 )(ε t-2 + θε t-3 )] = E[ε t ε t-3 + θε t-1 ε t-2 + θε t ε t-3 + θ 2 ε t-1 ε t-3 ] = E(ε t ε t-1 )+E(θε t-1 ε t-2 )+E(θε t ε t-3 )+E(θ 2 ε t-1 ε t-3 ) = =0  (  )=0 for all  >1 6.ρ(2) = Corr(y t,y t-2 ) =  (2)/  (0) = 0 ρ(  )=0 for all  >1

16 Population autocorrelation y t = ε t + 0.4ε t-1 ρ(0)=  (0)/  (0) = 1 ρ(1)=  (1)/  (0) = 0.4/( )=0.345 ρ(2)=  (2)/  (0) = 0/( ) = 0 ρ(  )=  (  )/  (0) = 0 for all  > 1

17 Population autocorrelation y t = ε t ε t-1 ρ(0)=  (0)/  (0) = 1 ρ(1)=  (1)/  (0) = /( )=0.499 ρ(2)=  (2)/  (0) = 0/( ) = 0 ρ(  )=  (  )/  (0) = 0 for all  > 1

18 MA(1): y t = ε t + θε t-1 [= (1+θL)ε t ] The partial autocorrelation function for the MA(1) process is a bit more tedious to derive. The PACF for an MA(1): The PACF, p(  ), will be nonzero for all , converging monotonically to zero in absolute value as  increases. If the MA coefficient θ is positive, the PACF will exhibit damped oscillations as  increases. If the MA coefficient θ is negative, then the PACF will be negative and converging to zero monotonically.

19 Population Partial Autocorrelation y t = ε t + 0.4ε t-1

20 Population Partial Autocorrelation y t = ε t ε t-1

21 Forecasting y T+h E(y T+h │ y T,y T-1,…) E(y T+h │ y T,y T-1,…) = E(y T+h │ ε T,ε T-1,…) since each y t can be expressed as a function of ε T,ε T-1,… E(y T+1 │ ε T,ε T-1,…) = E(ε T+1 + θε T │ ε T,ε T-1,…) since y T+1 = ε T+1 + θε T = E(ε T+1 │ ε T,ε T-1,…) + E(θε T │ ε T,ε T-1,…) = θε T E(y T+2 │ ε T,ε T-1,…)=E(ε T+2 + θε T+1 │ ε T,ε T-1,…) = E(ε T+2 │ ε T,ε T-1,…)+E(θε T+1 │ ε T,ε T-1,…) = 0 … E(y T+h │ y T,y T-1,…) = E(y T+h │ ε T,ε T-1,…) = θε T for h = 1 = 0 for h > 1

22 MA(q): 1.E(y t ) = 0 2.Var(y t ) = (1+b …+b q 2 )σ 2  (  ) and ρ(  ) will be equal to 0 for all  > q. [The behavior of these functions for 1 <  < q will depend on the signs and magnitudes of b 1,…,b q in a complicated way.] 4.The partial autocorrelation function, p(  ), will be nonzero for all . [Its behavior will depend on the signs and magnitudes of b 1,…,b q in a complicated way.]

23 MA(q): 5.E(y T+h │ y T,y T-1,…) = E(y T+h │ ε T, ε T-1,…) = ? y T+1 = ε T+1 + θ 1 ε T + θ 2 ε T-1 + … + θ q ε T-q+1 So, E(y T+1 │ ε T, ε T-1,…) = θ 1 ε T +θ 2 ε T-1 +…+ θ q ε T-q+1 More generally, E(y T+h │ ε T, ε T-1,…) = θ h ε T + … + θ q ε T-q+h for h < q 0 for h > q

24 Autoregressive Models (AR(p)) In certain circumstances, the Wold form for y t, can be “inverted” into a finite-order autoregressive form, i.e., y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t This is called a p-th order autoregressive process AR(p)). Note that it has p unknown coefficients: φ 1,…, φ p Note too that the AR(p) model looks like a standard linear regression model with zero-mean, homoskedastic, and serially uncorrelated errors.

25 AR(1): y t = φy t-1 + ε t

26 AR(1): y t = φy t-1 + ε t The “stationarity condition”: If y t is a stationary time series with an AR(1) form, then it must be that the AR coefficient, φ, is less than one in absolute value, i.e., │ φ │ < 1. To see how the AR(1) model is related to the Wold form – y t = φy t-1 + ε t = φ(φy t-2 + ε t-1 ) + ε t, since y t-1 = φy t-2 +ε t-1 = φ 2 y t-2 + φε t-1 + ε t = φ 2 (φy t-3 + ε t-2 ) + φε t-1 + ε t = φ 3 y t-3 + φ 2 ε t-2 + φε t-1 + ε t = (since │ φ │ < 1 and var(yt) <∞) So, the AR(1) model is appropriate for a covariance stationary process with Wold form

27 AR(1): y t = φy t-1 + ε t Mean of y t :E(y t ) = E(φy t-1 + ε t ) = φE(y t-1 ) + E(ε t ) = φE(y t ) + E(ε t ), by stationarity So, E(y t ) = E(ε t )/(1-φ) = 0, since ε t ~WN Variance of y t : Var(y t ) = E(y t 2 ) since E(y t ) = 0. E(y t 2 ) = E[(φy t-1 + ε t ) 2 ] = φ 2 E(y t 2 ) + E(ε t 2 ) + φE(y t-1 ε t ) (1- φ 2 )E(y t 2 ) = σ 2 E(y t 2 ) = σ 2 /(1- φ 2 )

28 AR(1): y t = φy t-1 + ε t  (1) =Cov(y t,y t-1 ) = E[(y t -E(y t ))(y t-1 -E(y t-1 ))] = E(y t y t-1 ) = E[ (φy t-1 + ε t )y t-1 ] = φE(y t-1 2 ) + E(ε t y t-1 ) = φE(y t-1 2 ) since E(ε t y t-1 ) = 0 = φ  (0), since E(y t-1 2 ) = Var(y t ) =  (0), ρ(1) = Corr(y t,y t-1 ) =  (1)/  (0) = φ > 0 if φ > 0 < 0 if φ < 0.

29 AR(1): y t = φy t-1 + ε t More generally, for the AR(1) process: ρ(  ) = φ  for all  So the ACF for the AR(1) process will Be nonzero for all values of , decreasing monotonically in absolute value to zero as  increases be strictly positive, decreasing monotonically to zero as  increases, if φ is positive alternate in sign as it decreases to zero, if φ is negative The PACF for an AR(1)will be equal to φ for  = 1 and will be equal to 0 otherwise, i.e., p(  ) = φ if  = 1 0 if  > 1

30 Population Autocorrelation Function AR(1): y t = 0.4y t-1 + ε t ρ(0)=  (0)/  (0) = 1 ρ(1)=  (1)/  (0) =φ=0.4 ρ(2)=  (2)/  (0) = φ 2 = 0.16 ρ(  )=  (  )/  (0) = φ  for all  > 1

31 Population Autocorrelation Function AR(1): y t = 0.95y t-1 + ε t ρ(0)=  (0)/  (0) = 1 ρ(1)=  (1)/  (0) =φ=0.95 ρ(2)=  (2)/  (0) = φ 2 = ρ(  )=  (  )/  (0) = φ  for all  > 1

32 Population Partial Autocorrelation Function AR(1): y t = 0.4y t-1 + ε t

33 Population Partial Autocorrelation Function AR(1): y t = 0.95y t-1 + ε t

34 AR(1): y t = φy t-1 + ε t E(y T+h │ y T,y T-1,…)= E(y T+h │ y T,y T-1,… ε T,ε T-1,…) 1. E(y T+1 │ y T,y T-1,…, ε T,ε T-1,…) = E(φy T +ε T+1 │ y T,y T-1,…, ε T,ε T-1,…) = E(φy T │ y T,y T-1,…, ε T,ε T-1,…) + E(ε T+1 │ y T,y T-1,…, ε T,ε T-1,…) = φy T 2. E(y T+2 │ y T,y T-1,…, ε T,ε T-1,…) = E(φy T+1 +ε T+2 │ y T,y T-1,…, ε T,ε T-1,…) = E(φy T+1 │ y T,y T-1,…, ε T,ε T-1,…) = φ E(y T+1 │ y T,y T-1,…, ε T,ε T-1,…) = φ(φy T ) = φ 2 y T 3. E(y T+h │ y T,y T-1,…) = φ h y T

35 Properties of the AR(p) Process y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t or, using the lag operator, φ(L)y t = ε t, φ(L) = 1- φ 1 L-…-φ p L p where the ε’s are WN(0,σ 2 ).

36 AR(p): y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t The coefficients of the AR(p) model of a covariance stationary time series must satisfy the stationarity condition: Consider the values of x that solve the equation 1-φ 1 x-…-φ p x p = 0 These x’s must all be greater than 1 in absolute value. For example, if p = 1 (the AR(1) case), consider the solutions to 1- φx = 0 The only value of x that satisfies this equation is x = 1/φ, which will be greater than one in absolute value if and only if the absolute value of φ is less than one. So, │ φ │ < 1 is the stationarity condition for the AR(1) model. The condition guarantees that the impact of ε t on y t+  decays to zero as  increases.

37 AR(p): y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t The autocovariance and autocorrelation functions,  (  ) and ρ(  ), will be non-zero for all . Their exact shapes will depend upon the signs and magnitudes of the AR coefficients, though we know that they will be decaying to zero as  goes to infinity. The partial autocorrelation function, p(  ), will be equal to 0 for all  > p. The exact shape of the pacf for 1 <  < p will depend on the signs and magnitudes of φ 1,…, φ p.

38 Population Autocorrelation Function AR(2): y t = 1.5y t y t-2 + ε t

39 AR(p): y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t E(y T+h │ y T,y T-1,…) = ? h = 1: y T+1 = φ 1 y T + φ 2 y T-1 +…+ φ p y T-p+1 +ε T+1 E(y T+1 │ y T,y T-1,…)=φ 1 y T +φ 2 y T-1 +…+φ p y T-p+1 h = 2: y T+2 = φ 1 y T+1 + φ 2 y T +…+ φ p y T-p+2 +ε T+2 E(y T+2 │ y T,y T-1,…) = φ 1 E(y T+1 │ y T,y T-1,…) + φ 2 y T +…+ φ p y T-p+2 h = 3 y T+3 = φ 1 y T+2 + φ 2 y T+1 + φ 3 y T +…+ φ p y T-p+3 +ε T+3 E(y T+3 │ y T,y T-1,…) = φ 1 E(y T+2 │ y T,y T-1,…) + φ 2 E(y T+1 │ y T,y T-1,…) + φ 3 y T +…+ φ p y T-p+3

40 AR(p): y t = φ 1 y t-1 + φ 2 y t-2 +…+ φ p y t-p +ε t E(y T+h │ y T,y T-1,…) = φ 1 E(y T+h-1 │ y T,y T-1,…) + φ 2 E(y T+h-2 │ y T,y T-1,…) +…+ φ p E(y T+h-p │ y T,y T-1,…) where E(y T+h-s │ y T,y T-1,…) = y T+h-s if h-s ≤0 In contrast to the MA(q), it is straightforward to operationalize this forecast. It is also straightforward to estimate this model: Apply OLS.

41 Planned exploratory regressions Series #1 of Problem Set #4 MA order 0123 AR order 0ARMA(0,0)ARMA(0,1)ARMA(0,2)ARMA(0,3) 1ARMA(1,0)ARMA(1,1)ARMA(1,2)ARMA(1,3) 2ARMA(2,0)ARMA(2,1)ARMA(2,2)ARMA(2,3) 3ARMA(3,0)ARMA(3,1)ARMA(3,2)ARMA(3,3) Want to find a regression model (the AR and MA orders in this case) such that the residuals look like white noise.

42 Model selection MA order 0123 AR order MA order 0123 AR order AIC SIC

43 ARMA(0,0): y t = c +  t The probability of observing the test statistics (Q-Stat) of under the null that the residual e(t) is white noise. That is, if e(t) is truly white noise, the probability of observing a test statistics of or higher is In this case, we will reject the null hypothesis.

44 ARMA(0,0): y t = c +  t The 95% confidence band for the autocorrelation under the null that residuals e(t) is white noise. That is, if e(t) is truly white noise, 95% of time (out of many realization of samples), the autocorrelation will fall within the band. We will reject the null hypothesis if the autocorrelation falls outside the band.

45 ARMA(0,0): y t = c +  t The PAC suggests AR(1).

46 ARMA(0,1)

47 ARMA(0,2)

48 ARMA(0,3)

49 ARMA(1,0)

50 ARMA(2,0)

51 ARMA(1,1)

52 AR or MA? ARMA(1,0) ARMA(0,3) Truth: y t = 0.5 y t-1 +  t We cannot reject the null that e(t) is white noise in both models.

53 Approximation Any MA process may be approximated by an AR(p) process, for sufficient large p. And the residuals will appear white noise. Any AR process may be approximated by a MA(q) process, for sufficient large q. And the residuals will appear white noise. In fact, if an AR(p) process can be written exactly as a MA(q) process, the AR(p) process is called invertible. Similarly, if a MA(q) process can be written exactly as an AR(p) process, the MA(q) process is called invertible.

54 Example: Employment MA(4) model

55 Residual plot

56 Correlogram of sample residual from an MA(4) model

57 Autocorrelation function of sample residual from an MA(4) model

58 Partial autocorrelation function of sample residual from an MA(4) model

59 Model AR(2)

60 Residual plot

61 Correlogram of sample residual from an AR(2) model

62 Model selection criteria – various MA and AR orders SIC values AIC values

63 Autocorrelation function of sample residual from an AR(2) model

64 Partial autocorrelation function of sample residual from an AR(2) model

65 ARMA(3,1)

66 Residual plot

67 Correlogram of sample residual from an ARMA(3,1) model

68 Autocorrelation function of sample residual from an ARMA(3,1) model

69 Partial autocorrelation function of sample residual from an ARMA(3,1) model

70 End