Parametric Estimation
In general, parametric models in regards to time series analysis refer to (mixed) autoregressive and moving average process models. Here, they are denoted in following way:
\[ y_{t} = \sum_{i=1}^{p} \phi_{i} y_{t-i} + \sum_{i=1}^{q} \theta_{i} \epsilon_{t-i} + \epsilon_{t},\]
where $\phi_{i}$ are the parameters of an autoregressive process of p-th order and $\theta_{i}$ are the parameters of an moving average process of q-th order and $\epsilon$ denotes the error term.
Most of the following functions are internally used by other estimators in this package, but because of their usefullness, they can be directly called by the user. Since for now wrapper functions are missing for each estimation method there exists a seperate function. For the special cases q = 0, use the ar functions and for p = 0 use maNR.
Autoregressiv Process AR(p)
TrendDecomposition.arBurg — Function
arBurg(y :: Vector, p :: Int; intercept::Bool = false)Fits an autoregressive model of order p to time series y using Burg's method.
Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.
TrendDecomposition.arOLS — Function
arOLS(y :: Vector, p :: Int; intercept::Bool = false)Fits an autoregressive model of order p to time series y using ordinary least square (OLS).
Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.
TrendDecomposition.arYuleWalker — Function
arYuleWalker(y :: Vector, p :: Int)Fits an autoregressive model of order p to time series y using the Yule-Walker estimators.
Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.
TrendDecomposition.arDurbinLevinson — Function
arDurbinLevinson(y :: Vector, p :: Int;
intercept::Bool = false, pacf::Bool = false)Fits an autoregressive model of order p to time series y using the Durbin-Levinson recursion.
Returns the tulpe (Φ, σ²), where Φ is the vector of estimated coefficients and σ² is the variance of the error terms.
Moving Average Process MA(p)
A moving average process, abbreviated as MA(q) can be estimated by maximum likelihood using the Newton-Raphson. This implementation follows Anderson[1], for that reason the parameters $\theta$ are denoted here as $\alpha$.
TrendDecomposition.maNR — Function
maNR(y :: Vector, q::Int = 1; iter::Int = 5, α = fill(0.2, q))Estimates a MA(q) model using the maximum likelihood function and the Newton-Raphson procedure.
Returns a tuple, (α, σ) where α is the (qx1) vector containing the MA coeffients, and σ is the estimated variance.
ARMA(p, q)
For now mixed autoregressive and moving average process, abbreviated as ARMA(p, q) can be estimated by maximum likelihood using the Newton-Raphson procedure. Both p and q have to be greater than 0. This estimation method was proposed by Åström and Torsten[2] and specifically this implementation follows the method proposed by Anderson[1], for that reason the parameter $\phi$ are denoted here as $\beta$ and the parameters $\theta$ as $\alpha$, as in the original paper. Normally, the algorithm converges very fast within 4 to 8 iterations, but sometimes the algorithm becomes unstable which could lead in this implementation to a singular matrix execption. In that case, choosing different starting points for the parameters can help to find a stable solution.
TrendDecomposition.armaNR — Function
armaNR(y :: Vector, p::Int = 1, q::Int = 1; iter::Int = 5, α = fill(0.4, q), β=[-1.])Estimates an ARMA(p, q) model using the maximum likelihood function and the Newton-Raphson procedure.
Both p and q have to be greater equal than one.
Returns a tuple, (β, α, σ) where β and α are the vectors containing the p AR and q MA coefficients respectively, and σ is the estimated variance.
- 1
Anderson, T. W. (1977). Estimation for autoregressive moving average models in the time and frequency domains. The Annals of Statistics, 5(5), 842-865.
- 2
Åström, K. J., & Torsten, B. (1965). Numerical identification of linear dynamic systems from normal operating records. IFAC Proceedings Volumes, 2(2), 96-111.