Partial Autocorrelation Function

For regression of y on x₁, x₂, x₃, x₄, the partial correlation between y and x₁ is

This can be calculated as the correlation between the residuals of the regression of y on x₂, x₃, x₄ with the residuals of x₁ on x₂, x₃, x₄.

For a time series, the h^th order partial autocorrelation is the partial correlation of y_i with y_i-h, conditional on y_i-1,…, y_i-h+1, i.e.

The first order partial autocorrelation is therefore the first-order autocorrelation.

The partial autocorrelations can be calculated as in the following alternative definition.

Definition 1: For k > 0, the partial autocorrelation function (PACF) of order k, denoted π_k, of a stochastic process, is defined as the k^th element in the column vector

Here, Γ_k is the k × k autocovariance matrix Γ_k = [v_ij] where v_ij = γ_|i-j| and δ_k is the k × 1 column vector δ_k = [γ_i]. We also define π₀ = 1. We can also define π_ki to be the i^th element in the vector , and so π_k = π_kk.

Provided γ₀ > 0, the partial correlation function of order k is equal to the k^th element in the following column matrix divided by γ₀.

Here, Σ_k is the k × k autocorrelation matrix Σ_k = [ω_ij] where ω_ij = ρ_|i-j| and τ_k is the k × 1 column vector τ_k = [ρ_i].

Note that if γ₀ > 0, then Σ_k and Γ_k are invertible for all m.

The partial autocorrelation function (PACF) of order k, denoted p_k, of a time series, is defined in a similar manner as the last element in the following matrix divided by r₀.

Here R_k is the k × k matrix R_k = [s_ij] where s_ij = r_|i-j| and C_k is the k × 1 column vector C_k = [r_i].

We also define p₀ = 1 and p_ik to be the i^th element in the matrix , and so p_k = p_kk. These values can also be calculated from the autocovariance matrix of the time series in the same manner as described above for stochastic processes.

Observation: Let Y be the k × 1 vector Y = [y₁ y₂ … y_k]^T and Z = [y₁–µ y₂–µ … y_k–µ]^T where µ = E[Y]. Then cov(Y) = E[ZZ^T], which is equal to the k × k autocovariance matrix Γ_k = [v_ij] where v_ij = γ_|i-j|.

Example 1: Calculate PACF for lags 1 to 7 for Example 2 of Autocorrelation Function.

The result is shown in Figure 1.

Figure 1 – PACF

We now show how to calculate PACF(4) in Figure 2. The calculations of the other PACF values is similar.

Figure 2 – Calculation of PACF(4)

First, we note that range R4:U7 of Figure 2 contains the autocovariance matrix with lag 4. This is a symmetric matrix, all of whose values come from range E4:E6 of Figure 1. The values on the main diagonal are s₀, the values on the diagonal above and below the main diagonal are s₁. The values on the diagonal two units away are s₂ and finally, the values in the upper right and lower left corners of the matrix are s₃.

Range R9:R12 is identical to range E5:E8. The values in range T9:T12 can now be calculated using the array formula

=MMULT(MINVERSE(R4:U7),R9:R12)

The values in range T9:T12 are the p_i4 values, and so we see that PACF(4) = p₄ = -.06685 (cell T12).

Observation: See Correlogram for information about the standard error and confidence intervals of the p_k, as well as how to create a PACF correlogram that includes the confidence intervals.

Observation: We see from the chart in Figure 1, that the PACF values for lags 2, 3, … are close to zero. As can be seen in Partial Autocorrelation for an AR(p) Process, this is typical for a time series derived from an autoregressive process. Note too that we can use Property 3 of Autocorrelation Function to test whether the PACF values for lags 2 and beyond are statistically equal to zero (see Figure 3).

Figure 3 – Bartlett’s test for PACF

The test shows that PACF(2) is not significantly different from zero. Note that, where applicable, we can also use Property 4 and 5 of Autocorrelation Function to test PACF values.

Real Statistics Function: The Real Statistics Resource Pack supplies the following function where R1 is a column range containing time series data:

PACF(R1, k) – the PACF value at lag k

The following array functions are also provided

ACOV(R1, k) – the autcovariance matrix at lag k

ACORR(R1, k) – the autcorrelation matrix at lag k

Property 1: The autocovariance matrix is non-negative definite.

Proof: Let Y = [y₁ y₂ … y_k]^T and let Γ be the k × k autocovariance matrix Γ = [v_ij] where v_ij = γ_|i-j| . As we observed previously, Γ = E[ZZ^T].

Now let X be any k × 1 vector. Then

which completes the proof.