Trends and seasonal components Flashcards
What is the classical decomposition model
Explains how time series is made up of linear combination of a trend component, a seasonal component and a stationary process (or noise) - we extract the trend and seasonal components to obtain a residual noise component which is stationary
Why do we require stationarity?
We always assume stationarity to be precise in estimating the autocorrelation as dependence is the most important feature to be captured in a time series.
Its difficult to measure dependence in a consistent way if the structure is not regular.
What are the different steps involved in eliminating mt (trend) and st(seasonal) components from Xt
1.Estimate mt and st - assume some form and estimate using ordinary least squares or regression etc
2. Smoothing
3. Differencing and transforming the time series: backshift operator, log transform, box cox etc
Assuming trend component takes the form of a polynomial how do we estimate the parameters in the trend?
Least squares estimation
What does a sample correlation with a steep decline indicate
Slow decay outside of proper bounds indicates no stationarity
How do we pick how many parameters to have in the trend component polynomial?
graph a realisation fo the series and see what shape it most reseembles - If its linear work with a linear trend etc
What is Xt - Mt called after detrending?
Residual
What assumptions do we make when approximating mt to the sample mean of Xt+j?
- mt is approximately linear on interval [t-q, t+q]
The average of the error terms over the intervla is close to 0
What is a problem using the weighted average approach to estimate mt?
Cannot use for t<q>n-q because this extends beyond observation. one solution is set Xt = X1 for all t<0 and Xt = Xn for all t>n</q>
How does differencing produce a staionary time series when Xt = mt +Yt
If mt is linear one differencing equation will turn time series into stationary one
If mt is quadratic two differencing turns are needed
etc
Polynomial of p - p differences applies then we will bet a stationary time series
What does the backshift operator do
Can move the time series back K units in time
What is a big positive of the differencing methods
No parameter estimation needed
When is data transformation useful
Often needed when dealing with non linear behaviour in observed time series - this approach is useful to stabalise variance, improve normal approximation and to improve linearity
What form will the seasonal component take
Often a periodic function
How can we estimate parameters in the mt and st component
Least squares method for example
