Provide Information Regarding Statistics & Econometrics : August 2025

Autocorrelation Function

Lecture 14

Autocorrelation Functions

(ACF)

Autocorrelation quantifies the linear relationship between a time series variable and a lagged value of itself over successive intervals. The autocorrelation diagnoses the time series characteristics and develops prediction models; it is essential to find patterns in the data, such as trends, seasonality, and stationarity or the absence thereof.

In other words,

Autocorrelation is a measure of how closely a time series resembles a lagged version of itself over a series of time intervals.

Let Yt, Yt-1, Yt-2, Yt-3, ..., Yt-k (t ∈ T, k = 1, 2, 3, ...) be the values of the time series phenomenon. Then the correlations between Yt and its lag Yt-k are denoted by rk and defined as:

Where:

Var(Yt) = Var(Yt-k)

Applications of ACF

i. ACF identifies seasonality in our time series data.

ii. ACF uncovered the hidden pattern in the time series data and helped the data scientist to select the correct method of forecasting.

iii. The analysis of ACF and PACF is necessary for selecting the appropriate ARIMA model for any time series.

Correlogram

(ACF Plot)

A correlogram is a visual way to show serial correlation in data change over time. In a correlogram or ACF plot, time lag is taken along the x-axis and autocorrelation along the y-axis.

The ACF can be used to determine a time series’ randomness and stationarity. In an ACF plot, each bar represents the size and direction of the connection. Bars that cross the red line are statistically significant.

Significance Bound

A significance bound is a confidence interval that indicates whether a spike is statistically significant or not. If the spikes fall outside of this range, they are considered statistically insignificant.

The significance bounds of ACF and PACF can be obtained as:

The spikes of ACF and PACF above +−Zα/2 / Sqrt n and below −Zα/2 / Sqrt n the bounds are significant.

Practice Question

Find the autocorrelation function of the following data. Also find 95 % confidence bounds

Time	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
	21	17	16	14	13	10	12	15	21	19	18	16	19	21	24	20

Solution:

Yt	Yt-1	Yt-2	Yt-3
21
17	21
16	17	21
14	16	17	21
13	14	16	17
10	13	14	16
12	10	13	14
15	12	10	13
21	15	12	10
19	21	15	12
18	19	21	15
16	18	19	21
19	16	18	19
20	19	16	18
24	21	19	16
21	24	20	19
276

Y¯t = ∑Yt / n

Y¯t = 276 / 16

Y¯t = 17.25


Yt - Y¯t	(Yt-1 - Y¯t)	(Yt-2 - Y¯t)	(Yt - Y¯t)(Yt-1 - Y¯t)	(Yt - Y¯t)^2
3.75
-0.25	3.75		-0.9375
-1.25	-0.25	3.75	0.3125
-3.25	-1.25	-0.25	4.0625
-4.25	-3.25	-1.25	13.8125
-7.25	-4.25	-3.25	30.8125
-5.25	-7.25	-4.25	38.0625
-2.25	-5.25	-7.25	11.8125
3.75	-2.25	-5.25	-8.4375
1.75	3.75	-2.25	6.5625
0.75	1.75	3.75	1.3125
-1.25	0.75	1.75	-0.9375
	-1.25	0.75	-2.1875
2.75	1.75	-1.25	4.8125
6.75	2.75	1.75	18.5625
3.75	6.75	2.75	25.3125
			142.9375	219

ACF(1)

ACF(2)

ACF(3)

ACF(4)

95% confidence bounds:1-α = 0.95α = 0.05Zα/2 = Z0.025 = 1.96

Partial Autocorrelation Function

(PACF)

The partial autocorrelation measures the relation between a variable at time t and lag version t – k while the linear effect of in-between lags is filter out Let Yt, Yt-1, Yt-2, Yt-3, ..., Yt-k be the values of the time series phenomenon. Then the partial auto-correlations between Yt and its lag Yt-k and the linear effect of Yt-1, Yt-2, Yt-3, ..., Yt-k-1 are filtered out.The 1^st order partial auto-correlation will be defined as:

The 1^st-order partial auto-correlation will be defined to equal the 1st-order auto-correlation.The 2^nd order (lag) partial auto-correlation is

The 3^rd order (lag) partial auto-correlation is

The k^th order (lag) partial auto-correlation is

Another way to compute PACF Let rt1, rt2, ..., rtk be sample auto-correlations of current time “t” and lag 1, 2, …, k. Then the partial auto correlation functions of time “t” and lag 1 while the effect of lag 2 is removed is defined as: