Nonlinear Time-Series Models¶

Table of Contents¶

• Linear Vs Nonlinear Adjustment
• Simple extension of the ARMA Model

Linear Vs Nonlinear Adjustment¶

Linear¶

VAR(p)

$$ Y_{t} = \pi_{1}Y_{t-1} +\pi_{2}Y_{t-2} + \dots+ \pi(Y_{t-p}) $$

In traditional econometrics, we have a conditional expectation on the variable of interest.

$$ E(Y_{t}|X_{t}) = \mu +a_{1}y_{t-1} + \dots + a_{k}Y_{t-k} $$

which is a linear regression as it is linear in parameters (= having degree 1).

$$ E(Y_{t}|Y_{t-i}) = \mu + \sum_{i=1}^k a_{i}Y_{t-i} + \epsilon_{t} $$

is a time series $(t = 1,2,3,\dots,T)$

$$ Y_{t} = \mu + \sum_{i=1}^k a_{i} X_{t} +\epsilon_{t} $$ in econometrics

The basic difference is that

• in econometrics, the average relationship is determined by both exogenous and endogenous variables.
• whereas in time series, the determinants are its own lagged values.

Non-linear¶

"Whatever is not linear is non-linear".

• The change in $X$ by 1 unit will not cause the same change in $Y$ at each level of $X$.
• Examples
  • MPC of the rich and poor differ (for food items)
  • Spending on luxury items for rich and poor is not the same.
• In such situations linearity may not give good forecasts. In fact, it would lead to high errors.

Thus, it is better to use non-linearity to improve forecasts.

But note that most of econometrics is based on the linear model.

• In financial time series, where variance relates to risk, using a linear model may not be appropriate.

$$ Y_{t} = f(Y_{t-1}, Y_{t-2},\dots,Y_{t-k}) + \epsilon_{t} $$

for $t = 1,2,\dots,T$. Each of the $Y_{t-i}$ are called arguments. If the model is linear in parameters then it is referred to as a linear model.

Since, non-linear models can take multitude of forms, it is often difficult to estimate a non-linear model.

On way to overcome this issue is that we choose the model depending on a situation, by imposing a restriction on its two moments, namely mean and variance.

$$ \begin{matrix} \mu_{t} & = E(y_{t}|F_{t-1}) & = g(\cdot,t) \\ \sigma^2_{t} & = V(y_{t}|F_{t-1}) & = h(\cdot,t) \\ \end{matrix} $$

By using the mean/variance function, $$ E(Y_{t}|F_{t-1}) + \sqrt{ h(\cdot,t)} \epsilon_{t} $$

• Note that here the non-linear error $\epsilon_{t}$ is multiplicative.

Evolution¶

1. 1980s (For economics and finance)
   • Bilinear
   • Threshold Autoregression
     • $y_{t} =\begin{cases}0.5 y_{t-1} + \epsilon_{t} & t \lt 2000 \\ -1 y_{t-1} + \eta_{t} & t \gt 2000 \\ \end{cases}$
       • Threshold $T = 2000$
     • Structural change is the essence of the TAR model
     • Example: For GDP (1960s - 2025), one equation is not sufficient
     • For multiple thresholds, use STAR
     • The threshold $T$ is chosen by the analysis
   • Smooth Autoregression (self excited TAR)
   • Markov Switching
     • First-order Markov chain: depends only on the current regime and not on the sequence of past events
     • Transition probability, $P_{ij}:$ Probability of jump from $i$-th to $j$-th state.
     • Switching from one regime (High) to another regime (Low) $\to$ Markov Switching
     • Use it when data exhibits structural change.
2. 1990s
   • Non Parametric
     • Distribution free
   • Semi Parametric
     • Some moments are known but not all
   • Parametric
     • Distribution is known $\implies$ All moments are known
3. Recent
   • AI
   • Neutral Networks
4. Current
   • AI/ML models

• Piece-wise linear: Non linear function for total period but consists of a combination of linear functions for different period with different slopes and intercepts.