# Chapter 22 Dynamic Causal Effects

## 22.1 Introduction

In the previous chapter, we look at a basic time series regression. Time Series appears in many different aspects of business. For example, projecting how much inventory to carry based off of past sales or how many sales to expect based on a history of advertising.

A draw back of regression is that it assumes everything happens simultaneously. However, this is not always the case. Take the advertising example, it may take a few weeks or months of advertising before we see the effects in sales.

Dynamic Causal inference is away of running a time series regression where \(Y_t\) is dependent not just on \(X_t\) but on past values of X \((X_{t-1},X_{t-2},...,X_{t-j})\).

We will only discuss the simplest model
\[Y_t=\beta_0+\beta_1 X_t+\beta_2 X_{t-1}+..\beta_j X_{t-1-j}+u_t\]
**Assumptions**
1. We need \(Y_t\) and \(X_t\) to be stationary.
2. We need \(X_t\) to be strictly exogenous.
3. We need \((Y_t,X_t)\) and \((Y_{t–j},X_{t–j})\) become independent as \(j\) gets large
4. We need no perfect multicolinearity.

## 22.2 Business Example

**Sales and Advertising**

```
library(modelsummary)
library(readr)
SALES <- read_table("https://www.econometrics.com/intro/SALES.txt", col_names = FALSE)
```

```
##
## ── Column specification ────────────────────────────────────────────────────────────────────────────
## cols(
## X1 = col_double(),
## X2 = col_double()
## )
```