代做ECO374H1 Moving Average (MA) Model Summer 2025代做留学生SQL语言

2. Moving Average (MA) Model

ECO374H1

Department of Economics

Summer 2025

White Noise Process

We will construct time series models out of building blocks

The simplest building block is the white noise process, {εt}, where each εt  is defined as an independent random shock with E [εt] = 0 and  Var (εt) = σε(2) for each t , and

ρk      =   0 for k ≥ 1

rk      =   0 for k ≥ 1

i.e.  both ACF and PACF are zero for all lags I  {εt} is a covariance stationary process

See R file 2a.  White Noise Simulation for simulated draws of the white noise process

Wold Decomposition Theorem

The white noise process is formally incorporated in linear time series models based on the following theorem, which implies that every covariance stationary stochastic process {Yt} can be written as the sum of two time series, one deterministic and one stochastic:

Theorem (Wold Decomposition Theorem)

If {Yt } is a covariance stationary process and {εt} is a white noise zero-mean process, then there exists a unique linear representation

where Vt is a deterministic component and is the stochastic component with ψ0 = 1, and .

Model Components

In the decomposition above, the sequence fεtg is called random shocks or innovations

Since , there must be a j from which all subsequent ψj+1, ψj+2, ... are getting smaller such that the corresponding innovations ε t -(j +1), ε t -(j +2), ... have a negligible effect on Yt

The deterministic component Vt can include a trend or cycle

In this Section we will assume Vt = 0, and return back to it with a full model in future Sections

Lag Operator Representation

We can write the Wold decomposition in terms of the lag operator L :

where we define the composite lag operator

Moving Average

We can approximate (1) with the model

called the Moving Average of order q, denoted by MA(q)

In practice, we will seek to have a good approximation of the dynamics in Yt with few parameters, for a small q