Document Type

Thesis

Degree Name

Master of Science (MSc)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Devan Becker

Advisor Role

Supervisor

Second Advisor

Mark Reesor

Advisor Role

Internal Examiner

Third Advisor

Wing Chan

Advisor Role

Internal-External Examiner

Abstract

Non-linearity in mediation analysis has been primarily studied in the context of binary variables which encode a treatment and control, estimating a `direct` and `indirect` effect of a covariate (a.k.a. treatment) X on a response Y, with a third variable M (the `mediator`) that is affected by X and in turn affects Y. The focus of mediation analysis in general is to quantify the overall effect of X on Y, including the effect through M. This presentation will focus on one or more continuous treatments and apply non-linear methods, including splines and machine learning models, to mediation analysis in a time series context.

We first explore splines for each of the models, leveraging the additive basis structure with consistent bases to define a mediation model with as many mediators as there are basis functions. The differentiability of splines allows us to use a derivative definition of the indirect and direct effect to determine how X affects Y.

We then explore ARX models, modeling X as an AR(p) model, and M, and Y as ARX models (with X and M as exogeneous variables in the definition of Y, and X as an exogenous variable in the definition of M), and find the associated covariances and autocovariances. With this structure, we note that the value of Y at time t depends on the value of X at time t-1 as well as M at time t-1, which in turn depends on X at time t-2; the effect of X on Y requires two time steps in order to go through M.

Finally, the derivative definition of the direct and indirect effects can be approximated numerically using the output of a black box model where necessary, allowing our results to apply a broader class of machine learning models. Overall, we provide a framework by which mediation analysis can be performed when X, M, and Y are all continuous time series random variables. The inspiration of this approach is for using economic indicators that have an assumed causal effect on each other.

Convocation Year

2027

Convocation Season

Spring

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