1 Midterm Review

2 Outline The Triangle of Stability - Power 7 The Triangle of Stability - Power 7 Augmented Dickey-Fuller Tests – Power 10 Augmented Dickey-Fuller Tests – Power 10 Lab Five Lab Five 2005 Midterm 2005 Midterm

3 I. The Triangle of Stability

4 Autoregressive Processes Order two: x(t) = b 1 * x(t-1) + b 2 * x(t-2) + wn(t) Order two: x(t) = b 1 * x(t-1) + b 2 * x(t-2) + wn(t) If b 2 = 0, then of order one: x(t) = b 1 * x(t-1) + wn(t) If b 2 = 0, then of order one: x(t) = b 1 * x(t-1) + wn(t) If b 1 = b 2 = 0, then of order zero: x(t) = wn(t) If b 1 = b 2 = 0, then of order zero: x(t) = wn(t)

5 Quadratic Form and Roots x(t) = b 1 * x(t-1) + b 2 * x(t-2) + wn(t) is a second order stochastic difference equation. If you drop the stochastic term, wn(t), then you have a second order deterministic difference equation: x(t) = b 1 * x(t-1) + b 2 * x(t-2) + wn(t) is a second order stochastic difference equation. If you drop the stochastic term, wn(t), then you have a second order deterministic difference equation: x(t) = b 1 * x(t-1) + b 2 * x(t-2), x(t) = b 1 * x(t-1) + b 2 * x(t-2), Or, x(t) - b 1 * x(t-1) - b 2 * x(t-2) = 0 Or, x(t) - b 1 * x(t-1) - b 2 * x(t-2) = 0 And substituting y 2-u for x(t-u), And substituting y 2-u for x(t-u), y 2 – b 1 *y – b 2 = 0, we have a quadratic equation y 2 – b 1 *y – b 2 = 0, we have a quadratic equation

6 Roots and stability Recall, for a first order autoregressive process, x(t) – b 1 *x(t-1)= wn(t), i.e. where b 2 = 0, then if we drop the stochastic term, and substitute y 1-u root is b 1. This root is unstable, i.e. a random walk if b 1 = 1. Recall, for a first order autoregressive process, x(t) – b 1 *x(t-1)= wn(t), i.e. where b 2 = 0, then if we drop the stochastic term, and substitute y 1-u for x(t-u), then y – b 1 =0, and the root is b 1. This root is unstable, i.e. a random walk if b 1 = 1. In an analogous fashion, for a second order process, if b 1 = 0, then x(t) = b 2 *x(t-2) + wn(t), similar to a first order process with a time interval twice as long, and if the root b 2 =1, we have a random walk and instability In an analogous fashion, for a second order process, if b 1 = 0, then x(t) = b 2 *x(t-2) + wn(t), similar to a first order process with a time interval twice as long, and if the root b 2 =1, we have a random walk and instability

7 Unit Roots and Instability Suppose one root is unity, i.e. we are on the boundary of instability, then factoring the quadratic from above, y 2 – b 1 *y – b 2 = (y -1)*(y –c), where (y- 1) is the root, y=1, and c is a constant. Multiplying out the right hand side and equating coefficients on the terms for y 2, y, and y 0 on the left hand side: Suppose one root is unity, i.e. we are on the boundary of instability, then factoring the quadratic from above, y 2 – b 1 *y – b 2 = (y -1)*(y –c), where (y- 1) is the root, y=1, and c is a constant. Multiplying out the right hand side and equating coefficients on the terms for y 2, y, and y 0 on the left hand side: y 2 – b 1 *y – b 2 = y 2 –y –c*y + c = y 2 –(1+c)*y + c y 2 – b 1 *y – b 2 = y 2 –y –c*y + c = y 2 –(1+c)*y + c So – b 1 = -(1+c), and –b 2 = c So – b 1 = -(1+c), and –b 2 = c Or b 1 + b 2 =1, and b 2 = 1 – b 1, the equation for a boundary of the triangle Or b 1 + b 2 =1, and b 2 = 1 – b 1, the equation for a boundary of the triangle

8 Triangle of Stable Parameter Space If b 2 = 0, then we have a first order process, ARTWO(t) = b 1 ARTWO(t-1) + WN(t), and for stability -1<b 1 <1 If b 2 = 0, then we have a first order process, ARTWO(t) = b 1 ARTWO(t-1) + WN(t), and for stability -1<b 1 <1 0 1 b1b1

9 Triangle of Stable Parameter Space If b 1 = 0, then we have a process, ARTWO(t) = b 2 ARTWO(t-2) + WN(t), that behaves like a first order process with the time interval two periods instead of one period. If b 1 = 0, then we have a process, ARTWO(t) = b 2 ARTWO(t-2) + WN(t), that behaves like a first order process with the time interval two periods instead of one period. For example, starting at ARTWO(0) at time zero, ARTWO(2) = b 2 ARTWO(0), ignoring the white noise shocks, and ARTWO(4) = b 2 ARTWO(2) = b 2 2 ARTWO(0) and for stability -1<b 2 <1 For example, starting at ARTWO(0) at time zero, ARTWO(2) = b 2 ARTWO(0), ignoring the white noise shocks, and ARTWO(4) = b 2 ARTWO(2) = b 2 2 ARTWO(0) and for stability -1<b 2 <1

10 Triangle of Stable Parameter Space b 1 = 0 b2b2 1

11 Triangle of Stable Parameter Space: b 1 = 0 b2b2 1 Draw a horizontal line through (0, -1) for (b 1, b 2 ) b 2 = 1 – b 1 If b 1 = 0, b 2 = 1, If b 1 = 1, b 2 =0, If b 2 = -1, b 1 =2 Boundary points +1

12 Triangle of Stable Parameter Space: b 1 = 0 b2b2 (0, 1) (0, -1) Draw a line from the vertex, for (b 1 =0, b 2 =1), though the end points for b 1, i.e. through (b 1 =1, b 2 = 0) and (b 1 =-1, b 2 =0), (1, 0)(-1, 0)

13 Triangle of Stable Parameter Space: b 1 = 0 b2b2 (0,1) (1, 0)(-1, 0) b 2 = 1 - b 1 Note: along the boundary, when b 1 = 0, b 2 = 1, when b 1 = 1, b 2 = 0, and when b 2 = -1, b 2 = 2. (2, -1)

14 Triangle of Stable Parameter Space: b 1 = 0 b2b2 (0, 1) (0, -1) (1, 0)(-1, 0) b 1 >0 b 2 >0 b 1 <0 b 2 >0 b 1 <0 b 2 <0 b 1 >0 b 2 <0

15 Is the behavior different in each Quadrant? b 1 = 0 b2b2 (0, 1) (0, -1) (1, 0)(-1, 0) b 1 >0 b 2 >0 b 1 <0 b 2 >0 b 1 <0 b 2 <0 b 1 >0 b 2 <0 I II III IV

16 We could study with simulation

17 Is the behavior different in each Quadrant? b 1 = 0 b2b2 (0, 1) (0, -1) (1, 0)(-1, 0) b 1 = 0.3 b 2 = 0.3 b 1 = -0.3 b 2 = 0.3 b 1 = -0.3 b 2 = -0.3 b 1 = 0.3 b 2 = -0.3 I II III IV

18 Simulation Sample Sample Genr wn=nrnd Genr wn=nrnd Sample 1 2 Sample 1 2 Genr artwo =wn Genr artwo =wn Sample Sample Genr artwo = 0.3*artwo(-1)+0.3*artwo(-2) + wn Genr artwo = 0.3*artwo(-1)+0.3*artwo(-2) + wn Sample Sample

19 II. Augmented Dickey-Fuller Tests

20 Part II: Unit Roots First Order Autoregressive or RandomWalk? First Order Autoregressive or RandomWalk? y(t) = b*y(t-1) + wn(t) y(t) = b*y(t-1) + wn(t) y(t) = y(t-1) + wn(t) y(t) = y(t-1) + wn(t)

21 Unit Roots y(t) = b*y(t-1) + wn(t) y(t) = b*y(t-1) + wn(t) we could test the null: b=1 against b<1 we could test the null: b=1 against b<1 instead, subtract y(t-1) from both sides: instead, subtract y(t-1) from both sides: y(t) - y(t-1) = b*y(t-1) - y(t-1) + wn(t) y(t) - y(t-1) = b*y(t-1) - y(t-1) + wn(t) or  y(t) = (b -1)*y(t-1) + wn(t) or  y(t) = (b -1)*y(t-1) + wn(t) so we could regress  y(t) on y(t-1) and test the coefficient for y(t-1), i.e. so we could regress  y(t) on y(t-1) and test the coefficient for y(t-1), i.e.  y(t) = g*y(t-1) + wn(t), where g = (b-1)  y(t) = g*y(t-1) + wn(t), where g = (b-1) test null: (b-1) = 0 against (b-1)< 0 test null: (b-1) = 0 against (b-1)< 0

22 Unit Roots i.e. test b=1 against b<1 i.e. test b=1 against b<1 This would be a simple t-test except for a problem. As b gets closer to one, the distribution of (b-1) is no longer distributed as Student’s t distribution This would be a simple t-test except for a problem. As b gets closer to one, the distribution of (b-1) is no longer distributed as Student’s t distribution Dickey and Fuller simulated many time series with b=0.99, for example, and looked at the distribution of the estimated coefficient Dickey and Fuller simulated many time series with b=0.99, for example, and looked at the distribution of the estimated coefficient

23 III. Lab Five

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35 Unit Roots Dickey and Fuller tabulated these simulated results into Tables Dickey and Fuller tabulated these simulated results into Tables In specifying Dickey-Fuller tests there are three formats: no constant-no trend, constant-no trend, and constant-trend, and three sets of tables. In specifying Dickey-Fuller tests there are three formats: no constant-no trend, constant-no trend, and constant-trend, and three sets of tables.

36 Unit Roots Example: the price of gold, weekly data, January 1992 through December 1999 Example: the price of gold, weekly data, January 1992 through December 1999

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44 Dickey-Fuller Tests The price of gold might vary around a “constant”, for example the marginal cost of production The price of gold might vary around a “constant”, for example the marginal cost of production P G (t) = MC G + RW(t) = MC G + WN(t)/[1-Z] P G (t) = MC G + RW(t) = MC G + WN(t)/[1-Z] P G (t) - MC G = RW(t) = WN(t)/[1-Z] P G (t) - MC G = RW(t) = WN(t)/[1-Z] [1-Z][P G (t) - MC G ] = WN(t) [1-Z][P G (t) - MC G ] = WN(t) [P G (t) - MC G ] - [P G (t-1) - MC G ] = WN(t) [P G (t) - MC G ] - [P G (t-1) - MC G ] = WN(t) [P G (t) - MC G ] = [P G (t-1) - MC G ] + WN(t) [P G (t) - MC G ] = [P G (t-1) - MC G ] + WN(t)

45 Dickey-Fuller Tests Or: [P G (t) - MC G ] = b* [P G (t-1) - MC G ] + WN(t) Or: [P G (t) - MC G ] = b* [P G (t-1) - MC G ] + WN(t) P G (t) = MC G + b* P G (t-1) - b*MC G + WN(t) P G (t) = MC G + b* P G (t-1) - b*MC G + WN(t) P G (t) = (1-b)*MC G + b* P G (t-1) + WN(t) P G (t) = (1-b)*MC G + b* P G (t-1) + WN(t) subtract P G (t-1) subtract P G (t-1) P G (t) - P G (t-1) = (1-b)*MC G + b* P G (t-1) - P G (t- 1) + WN(t) P G (t) - P G (t-1) = (1-b)*MC G + b* P G (t-1) - P G (t- 1) + WN(t) or  P G (t) = (1-b)*MC G + (1-b)* P G (t-1) + WN(t) or  P G (t) = (1-b)*MC G + (1-b)* P G (t-1) + WN(t) Now there is an intercept as well as a slope Now there is an intercept as well as a slope

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48 Augmented Dickey- Fuller Tests

49 ARTWO’s and Unit Roots Recall the edge of the triangle of stability: b 2 = 1 – b 1, so for stability b 1 + b 2 < 1 Recall the edge of the triangle of stability: b 2 = 1 – b 1, so for stability b 1 + b 2 < 1 x(t) = b 1 x(t-1) + b 2 x(t-2) + wn(t) x(t) = b 1 x(t-1) + b 2 x(t-2) + wn(t) Subtract x(t-1) from both sides Subtract x(t-1) from both sides x(t) – x(-1) = (b 1 – 1)x(t-1) + b 2 x(t-2) + wn(t) x(t) – x(-1) = (b 1 – 1)x(t-1) + b 2 x(t-2) + wn(t) Add and subtract b 2 x(t-1) from the right side: Add and subtract b 2 x(t-1) from the right side: x(t) – x(-1) = (b 1 + b 2 - 1) x(t-1) - b 2 [x(t-1) - x(t-2)] + wn(t) x(t) – x(-1) = (b 1 + b 2 - 1) x(t-1) - b 2 [x(t-1) - x(t-2)] + wn(t) Null hypothesis: (b 1 + b 2 - 1) = 0 Null hypothesis: (b 1 + b 2 - 1) = 0 Alternative hypothesis: (b 1 + b 2 -1)<0 Alternative hypothesis: (b 1 + b 2 -1)<0

50 Lab Five

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