Hello, everyone. In this lecture, we will estimate
model parameters of AR(3) Simulation. Objective is to estimate
model parameters of simulated AR(3) process using Yule-Walker
equations in matrix form. Exactly what we did in a previous lecture,
but this time it's not AR (2), it's AR (3). Now let's look at AR (3), not 2,
AR (3) process with mean 0. Now we have three terms instead of two. And in this simulation,
we're going to take p1 as 1 over 3, p2 as 1 over 2 and
p3 is going to be 7 over 100. And c1 is going to 4. Remember the Yule-Walker
equations in the matrix forum. In this case,
we'll be at three-by-three systems. This is our r matrix, this is our b,
the left hand side, and this is the coefficients that
we are trying to estimate. We will solve, we will find inverse of it,
multiply to the left. And remember Yule-Walker estimator for sigma square is basically
autocovariance at lag zero. 1 minus the dot product
of the coefficients and the autocorrelation onto lag p. In other words,
the sum of the pi ri, from 1 to p. Please go ahead and open up a notebook called AR(3)
simulation parameter estimation. And its name here in the notebook is
Yule-Walker Estimation AR(3) Simulation. So what are we doing? We are trying to estimate the model
parameters of an some simulations. So we have to simulate first,
this is just like before. We have set the c to 2017 so we can reproduce exactly what
we are producing this video. Sigma's four, this is our phi,
the coefficients. And this time we should
take 100,000 data points. Let's run the cell. Let's go to next code block. We are simulating AR(3) process,
again we are using arima.sim. And this time, we have three coefficients. So it automatically understands that this
is the autoregressive process of order 3. And n is 100,000. So we are simulating time series
with 100,000 data points in it. Let's run. So next one,
we define our autocorrelation, coefficients from 1 to 3, so r1, r2, r3. Okay, this is our r1,
this is r2, this is r3. And we define our matrix r and
then we update r accordingly so we get the r from
the Yule-Walker equation. And then r becomes basically once in
the main diagonal and it's very symmetric, is positive so we have a definite matrix. This is a base of r1s and this is r2. We define r matrix vector b
Basically by putting rs as a column. So this is our b, r1, r2, r3. And we solve rx = b and we are getting phi-hat here. And the phi-hat becomes basically 0.34,
0.33, almost 0.5 and this is almost like 0.07, which is very close to the actual model. And let's estimate sigma or variance. And the variance is around 15.97,
which is very close to 16. And next code block is basically
partitioning the output screen. And then putting time plot in
a CFN [INAUDIBLE] into three rows. If we do that, here we go. So this is our ar3 process,
100,000 data points. And this is autocorrelation
function which decays eventually. There's some exponential decay here. And partial autocorrelation function
has three significant correlations, lag1, lag2, lag3, and cuts off after lag3,
just like we expected. Results, we started with
100,000 data points. Phi1 is estimated at 0.338. This is phi2, this is phi3,
which is very close. The actual model is right here. This is where we started our
simulation and a fitted model is right here where innoations has
variance which is very close to 16. And this is our time plot ACF and PACF. So what have you learned? We have learned how to estimate
model parameters of ARP processes. This time it was AR(3) processes using
Yule-Walker equations in a matrix form. In the next lecture, we will actually look at the real time
series data [INAUDIBLE] simulation. And we'll try to fit autoregressive
processes into that dataset