国立中山大学：《计量经济学》（英文版） Chapter 12 Stochastic Process

Ch. 12 Stochastic Process 1 Introduction a particularly important aspect of real observable phenomena, which the random variables concept cannot accommodate, is their time dimension; the concept of random variable is essential static. A number of economic phenomena for which we need to formulate probability models come in the form of dynamic processes for which we have discrete sequence of observations in time. The problem we have to face is extend the simple probability model {f(x;6),6∈e}, to one which enables us to model dymamic phenomena. We have already moved in this direction by proposing the random vector probability model 6),6∈} The way w iewed this model so far has been as representing different char acteristics of the phenomenon in question in the form of the jointly distributed r.v.'s X1, X2, . XT. If we reinterpret this model as representing the same char acteristic but at successive points in time then this can be viewed as a dynamic probability model. With this as a starting point let us consider the dynamic probability model in the context of(S, F, P) 2 The Concept of a Stochastic Process its domain by attaching a date to the elements of the sample space to extend The natural way to make the concept of a random variable dynamic is Definition 1 Let(S, F, p)be a probability space and T an index set of real numbers and de- fine the function X(, by X(, : SxT-R. The ordered sequence of random variables X(, t),tET) is called a stochastic process This definition suggests that for a stochastic process ( X(, t),tET, for each tET, X(, t)represents a random variable on S. On the other hand, for each s

Ch. 12 Stochastic Process 1 Introduction A particularly important aspect of real observable phenomena, which the random variables concept cannot accommodate, is their time dimension; the concept of random variable is essential static. A number of economic phenomena for which we need to formulate probability models come in the form of dynamic processes for which we have discrete sequence of observations in time. The problem we have to face is extend the simple probability model, Φ = {f(x; θ), θ ∈ Θ}, to one which enables us to model dynamic phenomena. We have already moved in this direction by proposing the random vector probability model Φ = {f(x1, x2, ..., xT ; θ), θ ∈ Θ}. The way we viewed this model so far has been as representing different char￾acteristics of the phenomenon in question in the form of the jointly distributed r.v.’s X1, X2, ..., XT . If we reinterpret this model as representing the same char￾acteristic but at successive points in time then this can be viewed as a dynamic probability model. With this as a starting point let us consider the dynamic probability model in the context of (S, F,P). 2 The Concept of a Stochastic Process The natural way to make the concept of a random variable dynamic is to extend its domain by attaching a date to the elements of the sample space S. Definition 1: Let (S, F,P) be a probability space and T an index set of real numbers and de- fine the function X(·, ·) by X(·, ·) : S ×T → R. The ordered sequence of random variables {X(·,t),t ∈ T } is called a stochastic process. This definition suggests that for a stochastic process {X(·,t),t ∈ T }, for each t ∈ T , X(·,t) represents a random variable on S. On the other hand, for each s 1

in S, X(s, represents a function of t which we call a realization of the process X(s, t) for given s and t is just a real number Three main elements of a stochastic process (X(, t),tET are 1. its range space(sometimes called the state space), usually R; 2. the index z, usually one of R,R+=0,∞),z={…,0.,1,2,…}and 3. the dependence structure of the r v's X(, t),tETI In what follows a stochastic process will be denoted by IXt, teTi( s is dropped and X(t)is customary used as continuous stochastic process) and we are concerning exclusively on discrete stochastic process. That is, the index set T is a countable set such as T=0, 1, 2,. The dependence structure of (Xt, tET, in direct analogy with the case of a random vector, should be determined by the joint distribution of the process The question arises, however, since T is commonly an infinite set, do we need an infinite dimensional distribution to define the structure of the process This question was tackled by Kolmogorov(1933) who showed that when the stochastic process satisfies certain regularity conditions the answer is definitely no'. In particular, if we define the'tentative' joint distribution of the process for the subset(t1<t2<….<tr) of T by F(xh1,xt2,…,xtr)=Pr(Xn≤x1,Xt2≤ 2,.,Xtr <.r), then if the stochastic process (Xt, t E T satisfies the condi- tions 1. symmetry:F(xt1,x2,…,xtn)=F(xt1,xt2,…,x2r) where j1,j2,…,Tis any permutation of the indices 1, 2, ...,T(i.e. reshuffling the ordering of the in- dex does not change the distribution) 2. compatibility: limxr-oo F(ct,, Ita, Itr)= F(u, Tta,,tr_)(i.e. the dimensionality of the joint distribution can be reduced by marginalisation there exist a probability space(S, F, p) and a stochastic process Xt, tet de- IIn the function y= f(r), a is referred to as the argument of the function, and y is called the value of the function. We shall also alternatively refer z as the independent variable and y as the dependent variable. The set of all permissible value that a can take in a given context is known as the domain of the function. The value into which an r value is mapped is called he image of that a value. The set of all images is called the range of the function, which is the set of all values that y variable will take

in S, X(s, ·) represents a function of t which we call a realization of the process. X(s,t) for given s and t is just a real number. Three main elements of a stochastic process {X(·,t),t ∈ T } are: 1. its range space (sometimes called the state space),1 usually R; 2. the index T , usually one of R, R+ = [0, ∞), Z = {..., 0, 1, 2, ...} and 3. the dependence structure of the r.v.’s {X(·,t),t ∈ T }. In what follows a stochastic process will be denoted by {Xt ,t ∈ T } (s is dropped and X(t) is customary used as continuous stochastic process) and we are concerning exclusively on discrete stochastic process. That is, the index set T is a countable set such as T = {0, 1, 2, ...}. The dependence structure of {Xt ,t ∈ T }, in direct analogy with the case of a random vector, should be determined by the joint distribution of the process. The question arises, however, since T is commonly an infinite set, do we need an infinite dimensional distribution to define the structure of the process ? This question was tackled by Kolmogorov (1933) who showed that when the stochastic process satisfies certain regularity conditions the answer is definitely ’no’. In particular, if we define the ’tentative’ joint distribution of the process for the subset (t1 < t2 < ... < tT ) of T by F(xt1 , xt2 , ..., xtT ) = Pr(Xt1 ≤ x1, Xt2 ≤ x2, ..., XtT ≤ xT ), then if the stochastic process {Xt ,t ∈ T } satisfies the condi￾tions: 1. symmetry: F(xt1 , xt2 , ..., xtT ) = F(xtj1 , xtj2 , ..., xtjT ) where j1, j2, ..., jT is any permutation of the indices 1, 2, ..., T (i.e. reshuffling the ordering of the in￾dex does not change the distribution). 2. compatibility: limxT →∞ F(xt1 , xt2 , ..., xtT ) = F(xt1 , xt2 , ..., xtT −1 ) (i.e. the dimensionality of the joint distribution can be reduced by marginalisation); there exist a probability space (S, F,P) and a stochastic process {Xt ,t ∈ T } de- 1 In the function y = f(x), x is referred to as the argument of the function, and y is called the value of the function. We shall also alternatively refer x as the independent variable and y as the dependent variable. The set of all permissible value that x can take in a given context is known as the domain of the function. The value into which an x value is mapped is called the image of that x value. The set of all images is called the range of the function, which is the set of all values that y variable will take. 2

fined on it whose finite dimensional distribution is the distribution F(atu, Ct,., Ctr) as defined above. That is, the probability structure of the stochastic process IXt, te T is completely specified by the joint distribution of F(at, It2,.,tr) for all values of T(a positive integer) and any subset(t1, t2, . tr) of T Given that, for a specific t, Xt is a random variable, we can denote its distri- bution and density function by F(at) and f(t)respectively. Moreover the mean variance and higher moments of Xt(as a r v ) can be defined as standard form E(Xt) t E(Xt-Ht)2 (at-pt)2f(ct)drt =12(t),and E(Xt) allt∈T The linear dependence measures between Xt, and Xt (t,t)=E[(X4-p1)(X12一比),t,t∈T, is now called the autocovariance function. In standardized form r(ti, ti) v(ti)u t,t;∈T is called is autocorrelation function. These numerical characteristics of the stochastic process (Xt, te T) play an important role in the analysis of the pro- cess and its application to modeling real observable phenomena. We say that {Xt,t∈T}isan process if r(ti, ti)=0 t,t;∈T,t≠t Example One of the most important example of a stochastic process is the normal process The stochastic process (Xt, tE T is said to be normal (or Gaussian) if any finite subset of T, say t1, t2, .,tr, (Xt, X,,., Xt)=x] has a multivariate normal distribution, i.e (2)-/2IVrI-1 2 exp[-(xr -Hr)V7(xr-ur)

fined on it whose finite dimensional distribution is the distribution F(xt1 , xt2 , ..., xtT ) as defined above. That is, the probability structure of the stochastic process {Xt ,t ∈ T } is completely specified by the joint distribution of F(xt1 , xt2 , ..., xtT ) for all values of T (a positive integer) and any subset (t1,t2, ...,tT ) of T . Given that, for a specific t, Xt is a random variable, we can denote its distri￾bution and density function by F(xt) and f(xt) respectively. Moreover the mean, variance and higher moments of Xt (as a r.v.) can be defined as standard form as: E(Xt) = Z xt xtf(xt)dxt = µt , E(Xt − µt) 2 = Z xt (xt − µt) 2 f(xt)dxt = v 2 (t), and E(Xt) r = µrt , r ≥ 1, for all t ∈ T . The linear dependence measures between Xti and Xtj v(ti ,tj ) = E[(Xti − µti )(Xtj − µtj )], ti ,tj ∈ T , is now called the autocovariance function. In standardized form r(ti ,tj) = v(ti ,tj) v(ti)v(tj ) , ti ,tj ∈ T , is called is autocorrelation function. These numerical characteristics of the stochastic process {Xt ,t ∈ T } play an important role in the analysis of the pro￾cess and its application to modeling real observable phenomena. We say that {Xt ,t ∈ T } is an uncorrelated process if r(ti ,tj) = 0 for any ti ,tj ∈ T ,ti 6= tj . Example: One of the most important example of a stochastic process is the normal process. The stochastic process {Xt ,t ∈ T } is said to be normal (or Gaussian) if any finite subset of T , say t1,t2, ...,tT , (Xt1 , Xt2 , ..., XtT ) ≡ x 0 T has a multivariate normal distribution, i.e. f(xt1 , xt2 , ..., xtT ) = (2π) −T/2 |VT | −1/2 exp[− 1 2 (xT − µT ) 0V−1 T (xT − µT )], 3

where 1 22 v(t1, tr) U(t2,t)u2(t2) v(t2, tr) AT=e(xr)= u(tr, ti) t2(r) As in the case of a normal random variable. the distribution of a normal stochas- tic process is characterized by the first two moment but now they are function of t One problem so far in the definition of a stochastic process given above is much too general to enable us to obtain a operational probability model. In the analysis of stochastic process we only have a single realization of the process and we will have to deduce the value of At and v(t) with the help of a single bservation.(which is impossible The main purpose of the next three sections is to consider various special forms of stochastic process where we can construct probability models which are manageable in the context of statistical inference. Such manageability is achieved by imposing certain restrictions which enable us to reduce the number of unknown parameters involved in order to be able to deduce their value from a single real ization. These restrictions come in two forms: 1. restriction on the time-heterogeneity of the process; and 2. restriction on the memory of the process 2.1 Restricting the time-heterogeneity of a stochastic pro- cess For an arbitrary stochastic process Xt, tET) the distribution function F(at; et depends on t with the parameter 0. characterizing it being function of t as well That is, a stochastic process is time-heterogeneous in general. This, however raises very difficult issues in modeling real phenomena because usually we only have one observation for each t. Hence in practice we will have to estimate 0. on the basis of a single observation, which is impossible. For this reason we are going to consider an important class of stationary process which exhibit con- siderable time-homogeneity and can be used to model phenomena approaching

where µT = E(xT ) =         µ1 µ2 . . . µT         VT =         v 2 (t1) v(t1,t2) . . . v(t1,tT ) v(t2,t1) v 2 (t2) . . . v(t2,tT ) . . . . . . . . . . . . . . . . . . v(tT ,t1) . . . . v 2 (tT )         . As in the case of a normal random variable, the distribution of a normal stochas￾tic process is characterized by the first two moment but now they are function of t. One problem so far in the definition of a stochastic process given above is much too general to enable us to obtain a operational probability model. In the analysis of stochastic process we only have a single realization of the process and we will have to deduce the value of µt and v(t) with the help of a single observation. (which is impossible !) The main purpose of the next three sections is to consider various special forms of stochastic process where we can construct probability models which are manageable in the context of statistical inference. Such manageability is achieved by imposing certain restrictions which enable us to reduce the number of unknown parameters involved in order to be able to deduce their value from a single real￾ization. These restrictions come in two forms: 1. restriction on the time-heterogeneity of the process; and 2. restriction on the memory of the process. 2.1 Restricting the time-heterogeneity of a stochastic pro￾cess For an arbitrary stochastic process {Xt ,t ∈ T } the distribution function F(xt ; θt) depends on t with the parameter θt characterizing it being function of t as well. That is, a stochastic process is time-heterogeneous in general. This, however, raises very difficult issues in modeling real phenomena because usually we only have one observation for each t. Hence in practice we will have to estimate θt on the basis of a single observation, which is impossible. For this reason we are going to consider an important class of stationary process which exhibit con￾siderable time-homogeneity and can be used to model phenomena approaching 4

their equilibrium steady-state, but continuously undergoing random'func tions. This is the class of stationary stochastic processes Definition A stochastic process Xt, tET is said to be(strictly) stationary if any subset (t1, t2, ...,tr) of T and any T, 1+r;…tr+ That is, the distribution of the process remains unchanged when shifted in time by an arbitrary value T. In terms of the marginal distributions, (strictly)stationarity implies that F(X)=F(Xt+),t∈T, and hence F(t= F(t,)= F(rtr). That is stationarity implies that Xt, Xt, Xtr are(individually) identically distributed The concept of stationarity, although very useful in the context of probability theory, is very difficult to verify in practice because it is defined in terms of dis- tribution function. For this reason the concept of the second order stationarity defined in terms of the first two moments, is commonly preferred A stochastic process Xt, tET is said to be(weakly) stationary if E(Xt= u for all t; t4,t)=E(X4-)(x2-1)=m-4,t,t∈T These suggest that weakly stationarity for Xt, tE T) implies that its mean and variance u(ti)=7o are constant and free of t and its autocovariance depends on the interval t;-til; not ti and t;. Therefore, %k=)-k Example Consider the normal stochastic process in the above example. With the weakly

their equilibrium steady − state, but continuously undergoing ’random’ func￾tions. This is the class of stationary stochastic processes. Definition: A stochastic process {Xt ,t ∈ T } is said to be (strictly) stationary if any subset (t1,t2, ...,tT ) of T and any τ , F(xt1 , ..., xtT ) = F(xt1+τ , ..., xtT +τ ). That is, the distribution of the process remains unchanged when shifted in time by an arbitrary value τ . In terms of the marginal distributions, (strictly) stationarity implies that F(Xt) = F(Xt+τ ), t ∈ T , and hence F(xt1 ) = F(xt2 ) = ... = F(xtT ). That is stationarity implies that Xt1 , Xt2 , ..., XtT are (individually) identically distributed. The concept of stationarity, although very useful in the context of probability theory, is very difficult to verify in practice because it is defined in terms of dis￾tribution function. For this reason the concept of the second order stationarity, defined in terms of the first two moments, is commonly preferred. Definition: A stochastic process {Xt ,t ∈ T } is said to be (weakly) stationary if E(Xt) = µ for all t; v(ti ,tj) = E[(Xti − µ)(Xtj − µ)] = γ|tj−ti| , ti ,tj ∈ T . These suggest that weakly stationarity for {Xt ,t ∈ T } implies that its mean and variance v 2 (ti) = γ0 are constant and free of t and its autocovariance depends on the interval |tj − ti |; not ti and tj . Therefore, γk = γ−k. Example: Consider the normal stochastic process in the above example. With the weakly 5

stationarity assumption,now 71 ur= e(Xr) a sizeable reduction in the number of unknown parameters from T+T(T+1)/2 to(T+1). It is important, however, to note that even in the case of stationarity the number of parameters increase with the size of the subset(t1, . tr) although the parameters do not depend on t E T. This is because time-homogeneity does not restrict the 'memoryof the process. In the next section we are going to consider 'memory' restrictions in an obvious attempt to solve the problem of the parameters increasing with the size of the subset(t1, t2, .,tr) of T 2.2 Restricting the memory of a stochastic process n the case of a typical economic times series, viewed as a particular realization of a stochastic process (Xt, teT one would expect that the dependence between Xt, and X,, would tend to weaken as the distance(t-ti) increase. Formally, this dependence can be described in terms of the joint distribution F(at, Ita,. atr) as follows. Definition A stochastic process (Xt, tET is said to be asymptotically independent if for any subset(t1, t2,. tr) of and anv 7 B(r) defined by F(xt1,x2…,xr,xt1+,…,xxr+,)-F(xt1,c2…,x)F(x1+1…,xxr+ ≤B(7) goes to zero as T→∞. That is if B(7)→0asr→0 the two subsets(Xt1,X2,…,Xn)and(X1+,…,Xtr+) become independent a particular case of asymptotic independence is that of m-dependence which restricts B()to be zero for all T>m. That is, Xt and Xt, are independent for t1-t2|>m

stationarity assumption, now µT = E(XT ) =         µ µ . . . µ         VT =         γ0 γ1 . . . γT −1 γ1 γ0 . . . γT −2 . . . . . . . . . . . . . . . . . . γT −1 . . . . γ0         , a sizeable reduction in the number of unknown parameters from T +[T(T +1)/2] to (T + 1). It is important, however, to note that even in the case of stationarity the number of parameters increase with the size of the subset (t1, ...,tT ) although the parameters do not depend on t ∈ T . This is because time-homogeneity does not restrict the ’memory’ of the process. In the next section we are going to consider ’memory’ restrictions in an obvious attempt to ’solve’ the problem of the parameters increasing with the size of the subset (t1,t2, ...,tT ) of T . 2.2 Restricting the memory of a stochastic process In the case of a typical economic times series, viewed as a particular realization of a stochastic process {Xt , t ∈ T } one would expect that the dependence between Xti and Xtj would tend to weaken as the distance (tj −ti) increase. Formally, this dependence can be described in terms of the joint distribution F(xt1 , xt2 , ..., xtT ) as follows: Definition: A stochastic process {Xt , t ∈ T } is said to be asymptotically independent if for any subset (t1,t2, ...,tT ) of T and any τ , β(τ ) defined by |F(xt1 , xt2 , ..., xtT , xt1+τ , ..., xtT +τ ) − F(xt1 , xt2 , ..., xtT )F(xt1+τ , ..., xtT +τ )| ≤ β(τ ) goes to zero as τ → ∞. That is if β(τ ) → 0 as τ → ∞ the two subsets (Xt1 , Xt2 , ..., XtT ) and (Xt1+τ , ..., XtT +τ ) become independent. A particular case of asymptotic independence is that of m−dependence which restricts β(τ ) to be zero for all τ > m. That is, Xt1 and Xt2 are independent for |t1 − t2| > m. 6

Definition A stochastic process Xt, tET is said to be asymptotically uncorrelated if for there exists a sequence of constants p(T), T>l defined by u(tt+T (t)v(t+7) ≤p(r), for all t∈T such that 0≤p(7)≤1and∑p(r)1 defines an upper bound for the sequence of autocorrelation coefficients r(t, t+T). Moreover, given that p(r)→0asr→∞ is a necessary and p()0. a sufficient condition for 2-o p(r)1 define only upper bounds for the two measures of dependence given that when equality is used in their definition they will depend on(t1, t2,. tr) as well as T A more general formulation of asymptotic independence can be achieved using the concept of a a-field generated by a random vector. Let fi denote the a-field generated by X1, X2,., Xr where Xt, tE T is a stochastic process. A measure of the dependence among the elements of the stochastic process can be defined in terms of the events B∈ Et and A∈所rby a(T)=sup P(AnB)-P(A)P(B) munition A stochastic process Xt, tET) is said to be strongly mixing (a-miring) ifa(7)→0asr→∞

Definition: A stochastic process {Xt , t ∈ T } is said to be asymptotically uncorrelated if for there exists a sequence of constants {ρ(τ ), τ ≥ 1} defined by     v(t,t + τ ) v(t)v(t + τ )     ≤ ρ(τ ), for all t ∈ T , such that 0 ≤ ρ(τ ) ≤ 1 and X∞ τ=0 ρ(τ ) 0, a sufficient condition for P∞ τ=0 ρ(τ ) < ∞, the intuition underlying the above definition is obvious. At this stage it is important to note that the above concept of asymptotic independence and uncorrelatedness which restrict the memory of a stochastic process are not defined in terms of a stationary stochastic process but a general time-heterogeneous process. This is the reason why β(τ ) and ρ(τ ) for τ ≥ 1 define only upper bounds for the two measures of dependence given that when equality is used in their definition they will depend on (t1,t2, ...,tT ) as well as τ . A more general formulation of asymptotic independence can be achieved using the concept of a σ-field generated by a random vector. Let F t 1 denote the σ-field generated by X1, X2, ..., XT where {Xt ,t ∈ T } is a stochastic process. A measure of the dependence among the elements of the stochastic process can be defined in terms of the events B ∈ F t −∞ and A ∈ F ∞ t+τ by α(τ ) = sup τ |P(A ∩ B) − P(A)P(B)| . Definition: A stochastic process {Xt , t ∈ T } is said to be strongly mixing (α − mixing) if α(τ ) → 0 as τ → ∞. 7

A stronger form of mixing, sometimes called uniform miring, can be defined n terms of the following measure of dependence P(T)=sup P(AB)-P(A)L, P(B)>0 Definition: A stochastic process iXt, tET is said to be uniformly mixing (p-mi. ring) ifg(r)→0asr→∞ Looking at the two definitions of mixing we can see that a(r)and p(r)define absolute and relative measures of temporal dependence, respectively. The former based on the definition of dependence between two events A and B separate by T periods using the absolute measure P(A∩B)-P(A)·P(B≥0 and the latter the relative measure P(A|B)-P(A)≥0 Because p(r)>a(T)(why ) -mi ring implies a- miring In the context of weakly-stationary stochastic process, asymptotic uncorre- lateness can be defined more intuitively in terms of the temporal covariance as follows Cov(X,Xt+7)=7r→0 A stronger form of such memory restriction is so called ergodicity property Ergodicity can be viewed as a condition which ensures that the memory of the process as measured by weakens by averaging overtime Definitie A necessary condition is in the nature of a prerequisite: suppose that a statement

A stronger form of mixing, sometimes called uniform mixing, can be defined in terms of the following measure of dependence: ϕ(τ ) = sup τ |P(A|B) − P(A)| , P(B) > 0. Definition: A stochastic process {Xt , t ∈ T } is said to be uniformly mixing (ϕ − mixing) if ϕ(τ ) → 0 as τ → ∞. Looking at the two definitions of mixing we can see that α(τ ) and ϕ(τ ) define absolute and relative measures of temporal dependence, respectively. The former is based on the definition of dependence between two events A and B separated by τ periods using the absolute measure [P(A ∩ B) − P(A) · P(B)] ≥ 0 and the latter the relative measure [P(A|B) − P(A)] ≥ 0. Because ϕ(τ ) ≥ α(τ ) (why ?), ϕ − mixing implies α − mixing. In the context of weakly-stationary stochastic process, asymptotic uncorre￾latedness can be defined more intuitively in terms of the temporal covariance as follows: Cov(Xt , Xt+τ ) = γτ → 0 as τ → ∞. A stronger form of such memory restriction is so called ergodicity property. Ergodicity can be viewed as a condition which ensures that the memory of the process as measured by γτ ”weakens by averaging overtime” Definition: A necessary condition is in the nature of a prerequisite: suppose that a statement 8

p is true only if another statement q is true; then g constitutes a necessary condition of p. Symbolically, we express this as follows which is read 1."only if g, or alternative 2. if p, then g. It is also logically correct to mean 3.pimples g",and 4."pis a stronger condition than q"and 5. pc q Definition A weakly-stationary stochastic process iXt, tET) is said to be ergodic if ∑h< 3 Some special stochastic process We will consider briefly several special stochastic process which play an impor tant role in econometric modeling. These stochastic processes will be divided into parametric and non-parametric process. The non-parametric process are de- fined in terms of their joint distribution function or the first few joint moments On the other hand, parametric process are defined in terms of a generating mech. anism which is commonly a functional form based on a non-parametric process 3.1 Non-Parametric process Definition A stochastic process (Xt, tET is said to be a white-noise process if ().E(X) (ii E(X,X) 0ift≠ Hence, a white-noise process is both time-homogeneous, in view of the fact that it is a weakly-stationary process, and has no memory. In the case where (Xt, tETH is also assumed to be normal the process is also strictly stationary

p is true only if another statement q is true; then q constitutes a necessary condition of p. Symbolically, we express this as follows: p =⇒ q which is read: 1. ”p only if q, ” or alternative 2. ”if p, then q”. It is also logically correct to mean 3. ”p implies q”, and 4. ”p is a stronger condition than q ” and 5. p ⊂ q. Definition: A weakly-stationary stochastic process {Xt , t ∈ T } is said to be ergodic if X∞ τ=0 |γτ | < ∞. 3 Some special stochastic process We will consider briefly several special stochastic process which play an impor￾tant role in econometric modeling. These stochastic processes will be divided into parametric and non-parametric process. The non-parametric process are de- fined in terms of their joint distribution function or the first few joint moments. On the other hand, parametric process are defined in terms of a generating mech￾anism which is commonly a functional form based on a non-parametric process. 3.1 Non-Parametric process Definition: A stochastic process {Xt , t ∈ T } is said to be a white-noise process if (i). E(Xt) = 0; (ii). E(XtXτ ) =  σ 2 if t = τ 0 if t 6= τ. Hence, a white-noise process is both time-homogeneous, in view of the fact that it is a weakly-stationary process, and has no memory. In the case where {Xt , t ∈ T } is also assumed to be normal the process is also strictly stationary. 9

Definition A stochastic process iXt, tET is said to be a martingales process if Definition A stochastic process IXt, tET is said to be an innovation process if A stochastic process (Xt, tET) is said to be a Markov process if Definition: A stochastic process (Xt, tET) is said to be a Brownian motion process if 3.2 Parametric stochastic processes 3.2.1(Weakly) Stationary Process Definition A stochastic process (Xt, tET is said to be a autoregressive of order one (AR(1)) if it satisfies the stochastic difference equation, X where is a constant and ut is a white-noise process. We first consider the index setT*={0.,±1,±2,} and assume that X-r→0asT→∞ Define a lag -operator L by LXt≡Xt-1 then the ar(1) process can be rewritten as (1-OLXL when o| 1, it can be inverted as X2=(1-oD)-ut=(1+oL+2L2+…,)u=t2+0u-1+c2u1-2

Definition: A stochastic process {Xt , t ∈ T } is said to be a martingales process if... Definition: A stochastic process {Xt , t ∈ T } is said to be an innovation process if.... Definition: A stochastic process {Xt , t ∈ T } is said to be a Markov process if.... Definition: A stochastic process {Xt , t ∈ T } is said to be a Brownian motion process if... 3.2 Parametric stochastic processes 3.2.1 (Weakly) Stationary Process Definition: A stochastic process {Xt , t ∈ T } is said to be a autoregressive of order one (AR(1)) if it satisfies the stochastic difference equation, Xt = φXt−1 + ut where φ is a constant and ut is a white-noise process. We first consider the index set T ∗ = {0, ±1, ±2, ...} and assume that X−T → 0 as T → ∞. Define a lag − operator L by LXt ≡ Xt−1, then the AR(1) process can be rewritten as (1 − φL)Xt = ut , when |φ| < 1, it can be inverted as Xt = (1 − φL) −1 ut = (1 + φL + φ 2L 2 + ....)ut = ut + φut−1 + φ 2 ut−2 + ..... = X∞ i=0 φ iut−i , 10