The time series forecasting method that gives equal weightage to each of the M most recent observations is

The correct option is Moving average method.

Step-by-Step Explanation:

1. Understanding the Simple Moving Average (SMA):
The moving average method (specifically the Simple Moving Average) is a forecasting technique that estimates the value of a time series for the next period by taking the arithmetic mean of the $M$ most recent actual observations.

2. Mathematical Representation:
The forecast for period $t+1$, denoted as $F_{t+1}$, is calculated using the $M$ most recent actual values $Y_t,Y_{t-1},...,Y_{t-M+1}$ as follows:

$F_{t+1}=\frac{Y_t+Y_{t-1}+...+Y_{t-M+1}}{M}$

Alternatively, this can be written as:

$F_{t+1}=\frac{1}{M}{Y}_t+\frac{1}{M}{Y}_{t-1}+...+\frac{1}{M}{Y}_{t-M+1}$

3. Weightage Analysis:
From the equation above, we can see that each of the $M$ most recent observations is multiplied by a constant factor of $\frac{1}{M}$. This means that every observation within the sliding window of size $M$ is given equal weightage ($\frac{1}{M}$ or $\frac{100}{M}\%$), while any observation older than $M$ periods is assigned a weight of zero.

4. Comparison with other options:
- Exponential smoothing methods (including those with linear trend and triple exponential smoothing) assign weights that decrease exponentially as the observations get older.
- Kalman Filter uses a recursive Bayesian approach where weights are dynamically updated based on the estimated uncertainty and noise covariance, rather than assigning constant equal weights to a fixed historical window.