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There are several well-known one-factor interest rate models: Hull-White, Ho-Lee and Black-Derman-Toy just to name a few. There are also multi-factor models such as Longstaff-Schwartz and Chen. But what I haven't seen is multi-factor models with external variables, such as equity index volatility or GDP. Why don't these models exist--and if they do, what are they?

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These models do exist. They are known as "macro-finance" models. From "Macro-Finance Models of Interest Rates and the Economy":

During the past decade, much new research has combined elements of finance, monetary economics, and macroeconomics in order to study the relationship between the term structure of interest rates and the economy. In this survey, I describe three different strands of such interdisciplinary macro-finance term structure research. The first adds macroeconomic variables and structure to a canonical arbitrage-free finance representation of the yield curve. The second examines bond pricing and bond risk premiums in a canonical macroeconomic dynamic stochastic general equilibrium model. The third develops a new class of arbitrage-free term structure models that are empirically tractable and well suited to macro-finance investigations.

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