and 1()ttttyLLyεφφε=−⇒−=11Warwick Business School 10
Stationarityconditions for AR models
Note that the Woldrepresentation converges ifis the stationarityconditionfor an AR(1) process.An AR(p)process is defined:An AR(p) process is stationary if all the roots of the ‘characteristic equation’lie outsideof the unit circle. ()1 ,...12211<∞<+++=−=−−−φεφφεεεφtttttLy1<φ0...1221=−−−−ppzzzφφφ()()ttpptyLLLyLεφφφφ=−−−−≡...1221These roots ‘z’can be real or complex numbersWarwick Business School 11
Stationarityconditions: examples
The characteristic equation is:So this AR(1) process is stationary. Equivalently: Note that a random walk/martingaleis non-stationary-it’s an AR(1) process with 16.0106.01>=⇒=−zz()tttttyLyyεε=−+=−6.016.01process AR(1) Stationary16.0⇒<=φ1=φAR(1) processWarwick Business School 12
Stationarityconditions: examples
The characteristic equation is:⇒Note that the first differenceof y is a stationaryAR(1) process: ()()06.01106.06.112=−−=+−zzzz()ttttttyLLyyyεε=+−+−=−−2216.06.116.06.1()()LyyyLLttttt−≡Δ+Δ=Δ⇒=−−−16.016.011εεAR(2) process6.01 and 1==zzThis root means the process is non-stationaryExample: Recall that differencing (log) prices (a non-stationary process)results in a stationary series (log returns)Warwick Business School 13
Autocorrelation function (ACF) for AR(1) model
The ACF describes the ‘memory’of a stochastic process.For a stationary process the ACF will decay to zero.For a non-stationary process there is no decay.: ()()()224222523211222422220322112211...1...1.........φφσφφφσσφσφφσγφσσφσφσγεφφεεεφφεε−=+++=+++==−=+++==+++=+++=−−−−−−−tttttttttttyyEyEyy()1decay not does ingale walk/martrandom a of ACF The.1 ifdecay geometric infinitean has ACF,,Similarly.22011=<====φφφρφρφγγρkkKInfinite geometric series with:22 and φσ==raWarwick Business School 15
Moving average (MA) processes
Going back to the Woldrepresentation suppose⇒An MA(q) process is given byALLfinite order (q<∞) MA(q) models are stationary (the Woldform is convergent).However an important condition for MA models is invertibility. An MA(q) process is invertible if all the roots of lie outsideof the unit circle1 ,0 ,1>==jjψθψ()ttttLyεθθεε+=+=−11()()tqqttLLLLyεθθθεθ++++==...12210...1221=++++qqzzzθθθFirst order MA process: MA(1).Warwick Business School 16
Invertibility: example
The characteristic equation is:This MA(1) process is invertible. Invertibilitymeans that the process has a convergentinfinite order autoregressive representation15.0105.01−<−=⇒=+zz()ttttLyεεε5.015.01+=+=−()()ttttyLLLyLεε=+−+−=+−...125.025.05.015.013211<θThe directeffect of past observationsdecreases over time ⇒the AR form isconvergent. For the MA(1) process invertibilitymeans: Infinite geometric series withLra5.0 and 1−==Warwick Business School 17
ACF for MA models
For an MA(1) process the memory cuts off after the first lag: For an MA(q) process the memory cuts off (the auto-correlations are zero) after lag q.Again, this shows that all MA(q) (finite q) processes are stationary.()1 ,01121212222202111>=+==+=+=+=+=−−−−kyykttttttρθθρθσγσθσθσγθεεθεεACF of MA(1) process (theta=0.5)00.050.1
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