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 Time Series Forecasting:
In time series forecasting, we use historical data points to predict future values. Different forecasting methods assign different weights to past observations depending on their underlying mathematical formulations.

2. Analyzing the Moving Average Method:
The simple moving average (SMA) of order M calculates the average of the most recent M observations. Mathematically, the forecast at time t + 1, denoted by F_t+1, is given by:

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

We can rewrite this expression to show the weight associated with each observation:

$$F_{t+1}=\left(\frac{1}{M}\right)Y_t+\left(\frac{1}{M}\right)Y_{t-1}+\cdots +\left(\frac{1}{M}\right)Y_{t-M+1}$$

Here, each of the M most recent observations (from time t back to t - M + 1) is multiplied by the exact same weight,

$\frac{1}{M}$

Any observation older than M periods is given a weight of 0.

3. Comparing with Other Options:
- Exponential smoothing methods (including exponential smoothing with linear trend and triple exponential smoothing) assign exponentially decreasing weights to older observations, meaning recent data has much more weight than older data.
- Kalman Filter updates its predictions dynamically based on error covariance and does not apply equal static weights to the last M observations.

Thus, the method that assigns equal weightage to each of the M most recent observations is the Moving average method.