## 1. White noise

### 1.1. definition:

1.1.1. E(yt) = u

1.1.2. var(yt) = &2

1.1.3. COVRt-r =

1.1.3.1. &2 if t=r

1.1.3.2. 0 if otherwise

## 2. Stationary Processes

### 2.1. types

2.1.1. Strictly

2.1.1.1. A series is strictly stationary if the distribution of its values remains the same at time progresses

2.1.2. Weakly

2.1.2.1. E(yt) = u

2.1.2.2. E(yt-u)(yt-u)=&2 < 8

2.1.2.3. E(yt1-u)(yt2-u) = Rr2-t1 Vt1,t2

### 2.2. Stationary Time Series Models

2.2.1. A series is covariance stationary if its mean and all the covariances are unaffected by a change of time origin

2.2.2. Mean E(xt)=E(xt-s) = u

2.2.3. Variance var(xt) = E (xt - u)2 = E (xt-s - u)2 = &2

2.2.4. Covariance E[(xt - u)(xt-s - u)]= E[(xt-j - u)(xt-j-s - u)] = 9s

2.2.5. all constant

2.2.6. autocorrelation between xt and xt-s

## 3. Major Implications of Non-Stationarity

### 3.1. The stationarity or otherwise of a series can strongly influence its behaviour and properties - e.g. persistence of shocks will be infinite for nonstationary series

### 3.2. Spurious regressions. If two variables are trending over time, a regression of one on the other could have a high R2 even if the two are totally unrelated

### 3.3. If yt (the dependent variable) is not stationary, then it can be proved that the standard assumptions for asymptotic analysis will not be valid.

3.3.1. In other words, the usual “t-ratios” will not follow a t-distribution, so we cannot validly undertake hypothesis tests about the regression parameters.

## 4. Stationary/Non-Stationary Series

### 4.1. Stationary processes Characteristics

4.1.1. Shocks are temporary and hence dissipate over time

4.1.2. The series display mean reversion (i.e. it goes back to its long-run mean)

### 4.2. Non-stationary processes

4.2.1. Non-stationary processes characteristics

4.2.1.1. There is no long-run mean

4.2.1.2. The variance changes over time and is not bounded

4.2.1.3. The autocorrelations do not decay (do not rely on them to detect a unit root process)

4.2.2. By non-stationarity we mean the weak-form (covariance) non-stationarity.

4.2.3. 2 types

4.2.3.1. the random walk model with drift / stochastic non-stationarity

4.2.3.1.1. yt= u + yt-1 + ut

4.2.3.2. the deterministic trend process

4.2.3.2.1. yt = a + bt +ut

4.2.4. detrending a series

4.2.4.1. require different treatments to induce stationarity for 2 types

4.2.4.2. stochastic non-stationarity

4.2.4.2.1. ^yt=yt - yt-1

4.2.4.3. We will now concentrate on the stochastic non-stationarity model since deterministic non-stationarity does not adequately describe most series in economics or finance.

4.2.5. UNIT FOOT

4.2.5.1. definition

4.2.5.1.1. We can generalise this concept to consider the case where the series contains more than one “unit root”. That is, we would need to apply the first difference operator, , more than once to induce stationarity.

4.2.5.1.2. If a non-stationary series, yt must be differenced d times before it becomes stationary, then it is said to be integrated of order d. We write yt ~I(d).

4.2.5.2. test

4.2.5.2.1. The basic objective of the test is to test the null hypothesis that b =1 in: yt = byt-1 + ut

4.2.5.2.2. ^yt = 9yt-1 + ut

4.2.5.2.3. 3 types

4.2.5.2.4. ^yt = ut, test statistic = 9^ / SE(9)^

4.2.5.2.5. Augment Degree of Freedom

4.2.5.2.6. Testing for Integration of Higher Orders

4.2.5.2.7. Problems with the Unit Root Tests