The Autoregressive Model of Change

David A. Kenny

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Model• The present is determined by the past• X1 X2 X3 X4

• A mediational model: the relationship between two time points is explained by intermediate time points.– The relationship between X1 and X4

is explained either X2 or X3 and so r14.3 and r14.2 equal zero.

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Correlational Structure: Simplex

X1 1

X2 r1 1

X3 r1r2 r2 1

X4 r1r2r3 r2r3 r3 1

X1 X2 X3 X4

Generally the longer the lag, the weaker the correlation. Called a “simplex” correlational structure.

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Estimation• The first-order autoregressive model

can be estimated with as few as two waves of data.

• Model over-identified with three or more waves.

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Allowance for Measurement Error

• A much more realistic model, is a first-order autoregressive model with measurement error.

• Observed score equals true score plus error. The true score has an autoregressive structure.

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Identification• Partially identified with a least three waves.• Error variances and reliabilities of each wave

identified except for the first and last wave.• Autoregressive paths identified for each wave

except for wave 1 to wave 2. Standardized paths identified except for 1 to 2 and form the next to last to the last wave.

• Possible identifying assumption: equal error variances.

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Multiple Indicators• Need at least three indicators

per latent variable.• Correlate errors of the same

indicator at each time.• Need only two waves of data to

be identified.

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Second-Order Autoregressive

• A path from T1 to T3 (and T2 and T4).

• STARTS as a better conceptual alternative.